Use logarithmic differentiation: y = (x - 3)^3(x - 2)^5sqrt(x - 1)

(x - 3)^3(x - 2)^5sqrt(x - 1)((3)/(x - 3) + (5)/(x - 2) - (1)/(2(x - 1)))

Use logarithmic differentiation: y = (x + 2)^4(x - 3)^4sqrt(x - 2)

(x + 2)^4(x - 3)^4sqrt(x - 2)((4)/(x + 2) + (4)/(x - 3) - (1)/(2(x - 2)))

Use logarithmic differentiation: y = ((x - 1)^4(x - 3)^2)/((x + 4)^5)

((x - 1)^4(x - 3)^2)/((x + 4)^5)((4)/(x - 1) + (2)/(x - 3) - (5)/(x + 4))

Use logarithmic differentiation to find dy/dx for y = x^sin x

y' = x^sin x(cos xln x + (sin x)/(x))

Use logarithmic differentiation: y = ((x + 5)^3(x - 3)^5)/((x - 2)^2)

((x + 5)^3(x - 3)^5)/((x - 2)^2)((3)/(x + 5) + (5)/(x - 3) - (2)/(x - 2))

Use logarithmic differentiation to find dy/dx for y = (ln x)^x

y' = (ln x)^x(ln (ln x) + (1)/(ln x))

Use logarithmic differentiation: y = (x + 1)^4(x - 5)^2sqrt(x + 3)

(x + 1)^4(x - 5)^2sqrt(x + 3)((4)/(x + 1) + (2)/(x - 5) - (1)/(2(x + 3)))

Use logarithmic differentiation to find dy/dx for y = x^3sin x

y' = 3x^3sin x(cos xln x + (sin x)/(x))

Use logarithmic differentiation: y = ((x - 3)^2(x + 3)^4)/((x - 5)^4)

((x - 3)^2(x + 3)^4)/((x - 5)^4)((2)/(x - 3) + (4)/(x + 3) - (4)/(x - 5))

Use logarithmic differentiation to find dy/dx for y = (ln x)^2x

y' = 2(ln x)^2x(ln (ln x) + (1)/(ln x))

Use logarithmic differentiation: y = ((x + 4)^3)/((x + 1)^4)

((x + 4)^3)/((x + 1)^4)((3)/(x + 4) - (4)/(x + 1))

y = x^x + 1. Find y' by logarithmic differentiation.

y' = x^x + 1(ln x + (x + 1)/(x))

Worksheets · Calculus

Logarithmic differentiation worksheet

When x is in the exponent and in the base, neither the power rule nor the exponential rule applies. Take the natural log of both sides first: the exponent comes down as a factor, and the rest is the product rule.

Select your difficulty

Pick more than one for a sheet that mixes them.

The preview can look cramped on a phone. The PDF and printed sheet come out normal.



Sigma Prep · sigmaprep.io/worksheets Name Logarithmic Differentiation Date Period Use logarithmic differentiation. Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{e^{3 x}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 1$ for $y = \frac{{(x + 3)}^{2} {(x + 1)}^{2}}{x + 5}$ Use logarithmic differentiation: $y = \frac{{(x - 3)}^{3} {(x - 2)}^{5}}{\sqrt{x - 1}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 / x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = x^{4 x}$ Use logarithmic differentiation: $y = \frac{{(x + 2)}^{4} {(x - 3)}^{4}}{\sqrt{x - 2}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{4 \ln x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 0$ for $y = \frac{{(x + 3)}^{3} (x + 4)}{x + 1}$ Use logarithmic differentiation: $y = \frac{{(x - 1)}^{4} {(x - 3)}^{2}}{{(x + 4)}^{5}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{\sin x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x^{3} {(x + 1)}^{3}}{x + 5}$ Use logarithmic differentiation: $y = \frac{{(x + 5)}^{3} {(x - 3)}^{5}}{{(x - 2)}^{2}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 2)}^{3} (x - 1)}{x + 5}$ Use logarithmic differentiation: $y = \frac{{(x + 1)}^{4} {(x - 5)}^{2}}{\sqrt{x + 3}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 \sin x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x {(x + 1)}^{2}}{x + 4}$ Use logarithmic differentiation: $y = \frac{{(x - 3)}^{2} {(x + 3)}^{4}}{{(x - 5)}^{4}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{2 x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 3)}^{3} {(x - 1)}^{3}}{x + 5}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Logarithmic Differentiation

Date Period

Use logarithmic differentiation.

Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{e^{3 x}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 1$ for $y = \frac{{(x + 3)}^{2} {(x + 1)}^{2}}{x + 5}$
Use logarithmic differentiation:
$y = \frac{{(x - 3)}^{3} {(x - 2)}^{5}}{\sqrt{x - 1}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 / x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = x^{4 x}$
Use logarithmic differentiation:
$y = \frac{{(x + 2)}^{4} {(x - 3)}^{4}}{\sqrt{x - 2}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{4 \ln x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 0$ for $y = \frac{{(x + 3)}^{3} (x + 4)}{x + 1}$
Use logarithmic differentiation:
$y = \frac{{(x - 1)}^{4} {(x - 3)}^{2}}{{(x + 4)}^{5}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{\sin x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x^{3} {(x + 1)}^{3}}{x + 5}$
Use logarithmic differentiation:
$y = \frac{{(x + 5)}^{3} {(x - 3)}^{5}}{{(x - 2)}^{2}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 2)}^{3} (x - 1)}{x + 5}$
Use logarithmic differentiation:
$y = \frac{{(x + 1)}^{4} {(x - 5)}^{2}}{\sqrt{x + 3}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 \sin x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x {(x + 1)}^{2}}{x + 4}$
Use logarithmic differentiation:
$y = \frac{{(x - 3)}^{2} {(x + 3)}^{4}}{{(x - 5)}^{4}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{2 x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 3)}^{3} {(x - 1)}^{3}}{x + 5}$



Sigma Prep · sigmaprep.io/worksheets Name Logarithmic DifferentiationAnswer keyVersion 1 Date Period Use logarithmic differentiation. Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{e^{3 x}}$ $y^{'} = x^{e^{3 x}} e^{3 x} (3 \ln x + \frac{1}{x})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 1$ for $y = \frac{{(x + 3)}^{2} {(x + 1)}^{2}}{x + 5}$ $\frac{128}{9}$ Use logarithmic differentiation: $y = \frac{{(x - 3)}^{3} {(x - 2)}^{5}}{\sqrt{x - 1}}$ $\frac{{(x - 3)}^{3} {(x - 2)}^{5}}{\sqrt{x - 1}} (\frac{3}{x - 3} + \frac{5}{x - 2} - \frac{1}{2 (x - 1)})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 / x}$ $y^{'} = \frac{3 x^{3 / x} (1 - \ln x)}{x^{2}}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = x^{4 x}$ $1024 (\ln 2 + 1)$ Use logarithmic differentiation: $y = \frac{{(x + 2)}^{4} {(x - 3)}^{4}}{\sqrt{x - 2}}$ $\frac{{(x + 2)}^{4} {(x - 3)}^{4}}{\sqrt{x - 2}} (\frac{4}{x + 2} + \frac{4}{x - 3} - \frac{1}{2 (x - 2)})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{4 \ln x}$ $y^{'} = \frac{8 x^{4 \ln x} \ln x}{x}$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 0$ for $y = \frac{{(x + 3)}^{3} (x + 4)}{x + 1}$ $27$ Use logarithmic differentiation: $y = \frac{{(x - 1)}^{4} {(x - 3)}^{2}}{{(x + 4)}^{5}}$ $\frac{{(x - 1)}^{4} {(x - 3)}^{2}}{{(x + 4)}^{5}} (\frac{4}{x - 1} + \frac{2}{x - 3} - \frac{5}{x + 4})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{\sin x}$ $y^{'} = x^{\sin x} (\cos x \ln x + \frac{\sin x}{x})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x^{3} {(x + 1)}^{3}}{x + 5}$ $\frac{3564}{49}$ Use logarithmic differentiation: $y = \frac{{(x + 5)}^{3} {(x - 3)}^{5}}{{(x - 2)}^{2}}$ $\frac{{(x + 5)}^{3} {(x - 3)}^{5}}{{(x - 2)}^{2}} (\frac{3}{x + 5} + \frac{5}{x - 3} - \frac{2}{x - 2})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{x}$ $y^{'} = {(\ln x)}^{x} (\ln (\ln x) + \frac{1}{\ln x})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 2)}^{3} (x - 1)}{x + 5}$ $\frac{720}{49}$ Use logarithmic differentiation: $y = \frac{{(x + 1)}^{4} {(x - 5)}^{2}}{\sqrt{x + 3}}$ $\frac{{(x + 1)}^{4} {(x - 5)}^{2}}{\sqrt{x + 3}} (\frac{4}{x + 1} + \frac{2}{x - 5} - \frac{1}{2 (x + 3)})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 \sin x}$ $y^{'} = 3 x^{3 \sin x} (\cos x \ln x + \frac{\sin x}{x})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x {(x + 1)}^{2}}{x + 4}$ $3$ Use logarithmic differentiation: $y = \frac{{(x - 3)}^{2} {(x + 3)}^{4}}{{(x - 5)}^{4}}$ $\frac{{(x - 3)}^{2} {(x + 3)}^{4}}{{(x - 5)}^{4}} (\frac{2}{x - 3} + \frac{4}{x + 3} - \frac{4}{x - 5})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{2 x}$ $y^{'} = 2 {(\ln x)}^{2 x} (\ln (\ln x) + \frac{1}{\ln x})$ Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 3)}^{3} {(x - 1)}^{3}}{x + 5}$ $\frac{3025}{49}$ Free math worksheets at sigmaprep.io/worksheets© 2026 Sigma Prep

Sigma Prep · sigmaprep.io/worksheets

Name

Logarithmic DifferentiationAnswer keyVersion 1

Date Period

Use logarithmic differentiation.

Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{e^{3 x}}$ $y^{'} = x^{e^{3 x}} e^{3 x} (3 \ln x + \frac{1}{x})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 1$ for $y = \frac{{(x + 3)}^{2} {(x + 1)}^{2}}{x + 5}$ $\frac{128}{9}$
Use logarithmic differentiation:
$y = \frac{{(x - 3)}^{3} {(x - 2)}^{5}}{\sqrt{x - 1}}$ $\frac{{(x - 3)}^{3} {(x - 2)}^{5}}{\sqrt{x - 1}} (\frac{3}{x - 3} + \frac{5}{x - 2} - \frac{1}{2 (x - 1)})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 / x}$ $y^{'} = \frac{3 x^{3 / x} (1 - \ln x)}{x^{2}}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = x^{4 x}$ $1024 (\ln 2 + 1)$
Use logarithmic differentiation:
$y = \frac{{(x + 2)}^{4} {(x - 3)}^{4}}{\sqrt{x - 2}}$ $\frac{{(x + 2)}^{4} {(x - 3)}^{4}}{\sqrt{x - 2}} (\frac{4}{x + 2} + \frac{4}{x - 3} - \frac{1}{2 (x - 2)})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{4 \ln x}$ $y^{'} = \frac{8 x^{4 \ln x} \ln x}{x}$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 0$ for $y = \frac{{(x + 3)}^{3} (x + 4)}{x + 1}$ $27$
Use logarithmic differentiation:
$y = \frac{{(x - 1)}^{4} {(x - 3)}^{2}}{{(x + 4)}^{5}}$ $\frac{{(x - 1)}^{4} {(x - 3)}^{2}}{{(x + 4)}^{5}} (\frac{4}{x - 1} + \frac{2}{x - 3} - \frac{5}{x + 4})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{\sin x}$ $y^{'} = x^{\sin x} (\cos x \ln x + \frac{\sin x}{x})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x^{3} {(x + 1)}^{3}}{x + 5}$ $\frac{3564}{49}$
Use logarithmic differentiation:
$y = \frac{{(x + 5)}^{3} {(x - 3)}^{5}}{{(x - 2)}^{2}}$ $\frac{{(x + 5)}^{3} {(x - 3)}^{5}}{{(x - 2)}^{2}} (\frac{3}{x + 5} + \frac{5}{x - 3} - \frac{2}{x - 2})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{x}$ $y^{'} = {(\ln x)}^{x} (\ln (\ln x) + \frac{1}{\ln x})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 2)}^{3} (x - 1)}{x + 5}$ $\frac{720}{49}$
Use logarithmic differentiation:
$y = \frac{{(x + 1)}^{4} {(x - 5)}^{2}}{\sqrt{x + 3}}$ $\frac{{(x + 1)}^{4} {(x - 5)}^{2}}{\sqrt{x + 3}} (\frac{4}{x + 1} + \frac{2}{x - 5} - \frac{1}{2 (x + 3)})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{3 \sin x}$ $y^{'} = 3 x^{3 \sin x} (\cos x \ln x + \frac{\sin x}{x})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{x {(x + 1)}^{2}}{x + 4}$ $3$
Use logarithmic differentiation:
$y = \frac{{(x - 3)}^{2} {(x + 3)}^{4}}{{(x - 5)}^{4}}$ $\frac{{(x - 3)}^{2} {(x + 3)}^{4}}{{(x - 5)}^{4}} (\frac{2}{x - 3} + \frac{4}{x + 3} - \frac{4}{x - 5})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = {(\ln x)}^{2 x}$ $y^{'} = 2 {(\ln x)}^{2 x} (\ln (\ln x) + \frac{1}{\ln x})$
Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 2$ for $y = \frac{{(x + 3)}^{3} {(x - 1)}^{3}}{x + 5}$ $\frac{3025}{49}$

How to do these

Find $\frac{d y}{d x}$ for $y = x^{x}$

Take logs: ln y = x ln x.
Differentiate both sides: y'/y = ln x + 1.
Multiply by y: y' = x^x (ln x + 1).

The answer is $y^{'} = x^{x} (\ln x + 1)$ .

Where students go wrong

Using the power rule on x^x and writing x · x^(x - 1). The power rule needs a constant exponent, and here the exponent moves.

Also called log differentiation, differentiating x^x or derivative of x to the x.

What you can put on this worksheet

A variable to a variable power: x^x and others: Use logarithmic differentiation to find $\frac{d y}{d x}$ for $y = x^{x}$ → $y^{'} = x^{x} (\ln x + 1)$
The derivative at a point: Use logarithmic differentiation to find $\frac{d y}{d x}$ at $x = 1$ for $y = \frac{{(x + 4)}^{2} {(x + 2)}^{2}}{x + 5}$ → $\frac{135}{4}$
Products and quotients of powers and roots: Use logarithmic differentiation:
$y = \frac{{(x + 4)}^{3}}{{(x + 1)}^{4}}$ → $\frac{{(x + 4)}^{3}}{{(x + 1)}^{4}} (\frac{3}{x + 4} - \frac{4}{x + 1})$
Logarithmic differentiation: $y = x^{x + 1}$ . Find $y^{'}$ by logarithmic differentiation. → $y^{'} = x^{x + 1} (\ln x + \frac{x + 1}{x})$

Questions about these worksheets

Yes. Every logarithmic differentiation sheet is free to print and free to download as a PDF. There is no account to make, no email to hand over and no limit on how many you take.

Yes. The answer key prints on a second page, and you can choose whether it shows each question beside its answer or just the answers in a list.

Yes. Print straight from the page, or download the PDF and print that. The sheet is laid out for paper rather than squeezed off a screen, so there is room to work under each question, and the answer key comes out on its own page.

Calculus is usually taken around 12th grade or a first college course, though schools vary and plenty of students meet it earlier or later. Pick the difficulty rather than the grade: the easy level suits a first lesson on it, the hard level suits review before a test.

Yes, as many as you like. The questions are built fresh each time rather than picked from a fixed set of files, so pressing Generate gives a new sheet. That is what you want for a second class, a retake, or two students sitting next to each other.

That is the closest comparison, and Kuta Software is good software. It is also paid software you install on a computer. These worksheets run in a browser tab for free, with no account and nothing to install. Like Kuta, the logarithmic differentiation questions are generated when you ask for them rather than pulled from a fixed set of files, so two classes never get the same sheet, and the answer key comes with it.

Yes. There are three levels, and you can tick more than one for a sheet that mixes them, in which case the questions come out easiest first. The harder levels are not just larger numbers: they ask for something the easy ones do not.

You pick from 4: a variable to a variable power: x^x and others, the derivative at a point, products and quotients of powers and roots and logarithmic differentiation. Tick as many as you want and set how many of each, or let it spread them evenly.

Using the power rule on x^x and writing x · x^(x - 1). The power rule needs a constant exponent, and here the exponent moves.

Logarithmic differentiation worksheet

Select your difficulty

Logarithmic Differentiation

Logarithmic DifferentiationAnswer keyVersion 1

How to do these

Where students go wrong

What you can put on this worksheet

Questions about these worksheets

Are these logarithmic differentiation worksheets free?+

Do the logarithmic differentiation worksheets come with an answer key?+

Are these logarithmic differentiation worksheets printable?+

What grade are these logarithmic differentiation worksheets for?+

Can I get a different version of the same logarithmic differentiation worksheet?+

Is this a free alternative to Kuta Software for logarithmic differentiation?+

Can I choose how hard the logarithmic differentiation questions are?+

What kinds of question are on a logarithmic differentiation worksheet?+

What do students get wrong with logarithmic differentiation?+