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Exercises for Question Type 5: Applying quotient rule on functions as a rational - IB | RevisionDojo

Question 1

Find the derivative of $\frac{[\ln(x)]^2}{x^3}$ .

[4]

Question 2

Differentiate $\dfrac{e^{-2x}}{x^2+3}$ with respect to $x$ .

[4]

Question 3

Find the derivative of the function

$f(x) = \frac{\sqrt{1+x^2}}{x e^x}$

[6]

Question 4

Differentiate $\frac{\sin(x)+\ln(x)}{3x^2}$ with respect to $x$ .

[5]

Question 5

The question requires the application of the quotient rule, product rule, and chain rule to find the derivative of a complex transcendental function.

Find $\displaystyle \frac{d}{dx}\left[\frac{\cos(\ln x) e^{-x}}{x \ln x}\right]$ .

[5]

Question 6

Differentiate $\frac{\tan(x)}{x^2}$ with respect to $x$ .

[3]

Question 7

Differentiate $f(x) = \frac{\ln(x)}{\sqrt{x}+1}$ with respect to $x$ .

[5]

Question 8

The question requires students to differentiate a rational function using the quotient rule and simplify the resulting expression.

Differentiate $\frac{x^2+1}{x-1}$ with respect to $x$ .

[4]

Question 9

Differential calculus

Find the derivative of $f(x) = \dfrac{e^x + x^2}{\sin x}$ .

[3]

Question 10

Differentiate $\frac{\sin^2(x)}{\ln(x)}$ with respect to $x$ .

[3]

Question 11

The question tests differentiation of a composite function requiring the use of the quotient, product, and chain rules. Students must identify the correct rule applications and simplify their final result through factorization.

Find $\frac{d}{dx} \left[ \frac{x e^{x^2}}{\cos x} \right]$ .

[5]

Question 12

This question requires the application of differentiation rules, specifically the product and quotient rules.

Compute $\displaystyle \frac{d}{dx}\Bigl[\frac{x^3\cos(x)}{1+x^2}\Bigr]\,$ .

[4]

Question Type 5: Applying quotient rule on functions as a rational Exercises