Differentiation of trigonometric, logarithmic, exponential, composite and implicit functions — practice questions

Question 1 · difficulty L2 · understanding

Let $$f: \mathbb{R}-\{0\} \rightarrow \mathbb{R}$$ be a function satisfying $$f\left(\frac{x}{y}\right)=\frac{f(x)}{f(y)}$$ for all $$x, y, f(y) \neq 0$$. If $$f^{\prime}(1)=2024$$, then

• A. $$x f^{\prime}(x)+2024 f(x)=0$$
• B. $$x f^{\prime}(x)-2023 f(x)=0$$
• C. $$x f^{\prime}(x)-2024 f(x)=0$$
• D. $$x f^{\prime}(x)+f(x)=2024$$

Question 2 · difficulty L3 · understanding

Let f(x)=\sum_\limits{k=1}^{10} k x^{k}, x \in \mathbb{R}. If $$2 f(2)+f^{\prime}(2)=119(2)^{\mathrm{n}}+1$$ then $$\mathrm{n}$$ is equal to ___________

Question 3 · difficulty L3 · understanding

Let $$f^{1}(x)=\frac{3 x+2}{2 x+3}, x \in \mathbf{R}-\left\{\frac{-3}{2}\right\}$$ For $$\mathrm{n} \geq 2$$, define $$f^{\mathrm{n}}(x)=f^{1} \mathrm{o} f^{\mathrm{n}-1}(x)$$. If $$f^{5}(x)=\frac{\mathrm{a} x+\mathrm{b}}{\mathrm{b} x+\mathrm{a}}, \operatorname{gcd}(\mathrm{a}, \mathrm{b})=1$$, then $$\mathrm{a}+\mathrm{b}$$ is equal to ____________.

Question 4 · difficulty L3 · understanding

Let f : R $$\to$$ R satisfy $$f(x + y) = {2^x}f(y) + {4^y}f(x)$$, $$\forall$$x, y $$\in$$ R. If f(2) = 3, then $$14.\,{{f'(4)} \over {f'(2)}}$$ is equal to ____________.

Question 5 · difficulty L3 · understanding

Suppose for a differentiable function $$h, h(0)=0, h(1)=1$$ and $$h^{\prime}(0)=h^{\prime}(1)=2$$. If $$g(x)=h\left(\mathrm{e}^x\right) \mathrm{e}^{h(x)}$$, then $$g^{\prime}(0)$$ is equal to:

• A. 4
• B. 5
• C. 3
• D. 8

Question 6 · difficulty L3 · understanding

Let $$f(x)=x^5+2 \mathrm{e}^{x / 4}$$ for all $$x \in \mathbf{R}$$. Consider a function $$g(x)$$ such that $$(g \circ f)(x)=x$$ for all $$x \in \mathbf{R}$$. Then the value of $$8 g^{\prime}(2)$$ is :

• A. 4
• B. 2
• C. 16
• D. 8

Question 7 · difficulty L3 · understanding

$$\text { Let } y=\log _e\left(\frac{1-x^2}{1+x^2}\right),-1 < x<1 \text {. Then at } x=\frac{1}{2} \text {, the value of } 225\left(y^{\prime}-y^{\prime \prime}\right) \text { is equal to }$$

• A. 732
• B. 736
• C. 742
• D. 746

Question 8 · difficulty L3 · understanding

Let $$f(x)=\frac{\sin x+\cos x-\sqrt{2}}{\sin x-\cos x}, x \in[0, \pi]-\left\{\frac{\pi}{4}\right\}$$. Then $$f\left(\frac{7 \pi}{12}\right) f^{\prime \prime}\left(\frac{7 \pi}{12}\right)$$ is equal to

• A. $$\frac{2}{3 \sqrt{3}}$$
• B. $$\frac{2}{9}$$
• C. $$\frac{-1}{3 \sqrt{3}}$$
• D. $$\frac{-2}{3}$$

Question 9 · difficulty L3 · understanding

If $$2 x^{y}+3 y^{x}=20$$, then $$\frac{d y}{d x}$$ at $$(2,2)$$ is equal to :

• A. $$-\left(\frac{3+\log _{e} 16}{4+\log _{e} 8}\right)$$
• B. $$-\left(\frac{2+\log _{e} 8}{3+\log _{e} 4}\right)$$
• C. $$-\left(\frac{3+\log _{e} 8}{2+\log _{e} 4}\right)$$
• D. $$-\left(\frac{3+\log _{e} 4}{2+\log _{e} 8}\right)$$

Question 10 · difficulty L3 · understanding

If $$y(x)=x^{x},x > 0$$, then $$y''(2)-2y'(2)$$ is equal to

• A. $$4(\log_{e}2)^{2}+2$$
• B. $$8\log_{e}2-2$$
• C. $$4\log_{e}2+2$$
• D. $$4(\log_{e}2)^{2}-2$$