COMPLEX QUESTIONS ON DIFFERENTIATION

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Differentiation is the process of finding the derivative of a function. 

So far, we have looked at different aspects of differentiation such as differentiation of logarithm, natural logarithmic, differentiation via quotient rule and chain rule.

To continue our series on differentiation, we will attempt to solve some complex differentiation problems.

Forthwith, let's get started.

Question 1

Differentiate the function $y=\sin (\ln(x^4+2))$

Answer

Let $u=\ln(x^4+2)$, so that

$y=\sin (u)$ and

$\frac{dy}{du}=\cos u$ and $\frac{du}{dx}=\frac{4x^3}{x^4+2}$

Recall that the derivative of a composite function is

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$

Accordingly

$$\frac{dy}{dx}=\cos u \times\frac{4x^3}{x^4+2}$$

$$\frac{dy}{dx}=\frac{\cos u \times 4x^3}{x^4+2}$$

$$\frac{dy}{dx}=\frac{4x^3\cos u}{x^4+2}$$

Because $u=\ln(x^4+3)$

$$\frac{dy}{dx}=\frac{4x^3\cos (\ln(x^4+3))}{x^4+2}$$

Question 2

If $h(x)=\sin (e^{7x})$, Find h'(x)

Answer

Let $u=e^{7x}$ so that 

$\frac{dy}{du}=\cos u$ and 

$\frac{du}{dx}=7e^{7x}$

Since the derivative of a composite function is

$$\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$$

Therefore, 

$$\frac{dy}{dx}=\cos u\times 7e^{7x}$$

Because $u=e^{7x}$

$$\frac{dy}{dx}=\cos (e^{7x})\times 7e^{7x}$$

Question 3

Find the derivative if $y=3^{\ln(x+10)}$

Answer

The derivative of an exponential function is usually given by the expression

$$\frac{d}{dx}a^{f(x)}=a^{f(x)}\times \ln(a)\times f'(x)$$

Using that same logic

$$\frac{dy}{dx}=3^{\ln(x+10)}\times \ln(3)\times \frac{1}{x+10}$$

$$\frac{dy}{dx}=\frac{\ln(3) \times 3^{\ln(x+10)}}{x+10}$$

Question 4

Differentiate $y=2^xe^{5x}$

Answer

$2^xe^{5x}$ is the product of $2^x$ and $e^{5x}$. Hence, we simply use the product rule:

$(f\times g)=f'g+g'f$

Here, $f=2^x$, $g=e^{5x}$, $f'=\ln (2)\times 2^{x}$ and $g'=5e^{5x}$

$$\frac{dy}{dx}=(\ln (2)\times 2^{x})(e^{5x})+(5e^{5x})(2^x)$$

$$\frac{dy}{dx}=\ln (2)2^{x}e^{5x}+2^x5e^{5x}$$

By factorization, 

$$\frac{dy}{dx}=e^{5x}2^x(\ln (2) +5)$$

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Question 5

Find the derivative of $y=-\sin 3x+2x+(x+9)^5+3$

Answer

There isn't much to solve here; simply apply the sum and difference rule of differentiation.

$$\frac{dy}{dx}=-3\cos 3x+2x+5(x+9)^4$$

Question 6

Obtain the derivative of $e^{\cos x}$

Solution

Recalled that the derivative of

$$e^{a} ×a^1$$

Accordingly,

$f'(e^{\cos x})=e^{\cos x} × -\sin x$

$f'(e^{\cos x})=-e^{\cos x}\sin x$.

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