NZ Level 8 (NZC) Level 3 (NCEA) [In development]
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Differentiation various functions
Lesson

In routine problems requiring differentiation, we rely on several previously established rules for differentiating various types of functions. However, one should keep in mind the meaning of the derivative as the function that gives the rate of change or gradient of the original function at each point in the domain.

Specifically, if $f$f is a smoothly continuous function, then its derivative at a number $x$x in the domain is

$f'(x)=\lim_{a\rightarrow x}\frac{f(x)-f(a)}{x-a}$f(x)=limaxf(x)f(a)xa

The rules that are used in differentiation have been deduced from and are consistent with this definition.

 

In practice, we make much use of the facts that

  • The derivative of a sum is the sum of the separate derivatives.
  • The derivative of a constant multiple of a function is the constant multiple multiplied by the derivative of the function.

We often need the

  • product rule: If $f(x)=g(x)\cdot h(x)$f(x)=g(x)·h(x), then $f'(x)=g'(x)\cdot h(x)+g(x)\cdot h'(x)$f(x)=g(x)·h(x)+g(x)·h(x)
  • quotient rule: If $f(x)=\frac{g(x)}{h(x)}$f(x)=g(x)h(x), then $f'(x)=\frac{g'(x)\cdot h(x)-g(x)\cdot h'(x)}{\left[h(x)\right]^2}$f(x)=g(x)·h(x)g(x)·h(x)[h(x)]2
  • function-of-a-function rule (also called the chain rule): If $f(x)=g\left(h(x)\right)$f(x)=g(h(x)), then $f'(x)=g'\left(h(x)\right)\cdot h'(x)$f(x)=g(h(x))·h(x), provided $h(x)$h(x) is in the domain of $g(x)$g(x)

 

When the function-of-a-function rule is expressed in the Leibniz notation, it is possible to see the reason for the term chain rule. If, for example, a function is given by $u\left(v(x)\right)$u(v(x)), we write $\frac{\mathrm{d}u}{\mathrm{d}x}=\frac{\mathrm{d}u}{\mathrm{d}v}\cdot\frac{\mathrm{d}v}{\mathrm{d}x}$dudx=dudv·dvdx

 

We may also need the derivatives of some special functions.

  • $\frac{\mathrm{d}}{\mathrm{d}x}\ln x=\frac{1}{x}$ddxlnx=1x
  • $\frac{\mathrm{d}}{\mathrm{d}x}\sin x=\cos x$ddxsinx=cosx
  • $\frac{\mathrm{d}}{\mathrm{d}x}\cos x=-\sin x$ddxcosx=sinx
  • $\frac{\mathrm{d}}{\mathrm{d}x}e^x=e^x$ddxex=ex

 

Example 1

Find the derivative of $\sin\left(\ln x\right)$sin(lnx).

We need the function-of-a-function rule. It may be helpful to think of an inside function $\ln$ln and an outside function $\sin$sin. We differentiate the outside function first, evaluated at $\ln x$lnx, and multiply by the derivative of the inside function. Hence, if $f(x)=\sin\left(\ln x\right)$f(x)=sin(lnx), we have

$f'(x)=\cos\left(\ln x\right)\cdot\left(\frac{1}{x}\right)$f(x)=cos(lnx)·(1x)

 

Example 2

Find the derivative of the function $f(x)=\left(x+\sin x\right)^{\frac{3}{2}}$f(x)=(x+sinx)32 and calculate the gradient of $f$f at $x=0$x=0, $x=\frac{\pi}{4}$x=π4 and $x=\frac{\pi}{2}$x=π2.

First, we differentiate the power function to obtain $\frac{3}{2}\left(x+\sin x\right)^{\frac{1}{2}}$32(x+sinx)12. Then, we differentiate the inside function to obtain $1+\cos x$1+cosx. Finally, we have

$f'(x)=\frac{3}{2}\left(x+\sin x\right)^{\frac{1}{2}}\cdot\left(1+\cos x\right)$f(x)=32(x+sinx)12·(1+cosx).

We calculate, $f'(0)=\frac{3}{2}\left(0+\sin0\right)^{\frac{1}{2}}\cdot\left(1+\cos0\right)=0$f(0)=32(0+sin0)12·(1+cos0)=0,

Next, $f'\left(\frac{\pi}{4}\right)=\frac{3}{2}\left(\frac{\pi}{4}+\sin\frac{\pi}{4}\right)^{\frac{1}{2}}\cdot\left(1+\cos\frac{\pi}{4}\right)=\frac{3}{2}\sqrt{\frac{\pi}{4}+\frac{1}{\sqrt{2}}}\cdot(1+\frac{1}{\sqrt{2}})\approx3.128$f(π4)=32(π4+sinπ4)12·(1+cosπ4)=32π4+12·(1+12)3.128

And lastly, $f'\left(\frac{\pi}{2}\right)=\frac{3}{2}\left(\frac{\pi}{2}+\sin\frac{\pi}{2}\right)^{\frac{1}{2}}\cdot\left(1+\cos\frac{\pi}{2}\right)=\frac{3}{2}\sqrt{\frac{\pi}{2}+1}\approx2.405$f(π2)=32(π2+sinπ2)12·(1+cosπ2)=32π2+12.405

 

Example 3

Find the gradient of $f(x)=e^{-x}\left(x^3-1\right)$f(x)=ex(x31) at $x=1$x=1.

Using the product rule and the function-of-a-function rule, we have $f'(x)=(-1)e^{-x}\left(x^3-1\right)+e^{-x}\left(3x^2\right)$f(x)=(1)ex(x31)+ex(3x2)

This is written more tidily as $f'(x)=e^{-x}\left(1+3x^2-x^3\right)$f(x)=ex(1+3x2x3).

Then, $f'(1)=e^{-1}\times3=\frac{3}{e}\approx1.1$f(1)=e1×3=3e1.1.

 

Worked Examples

Question 1

Differentiate $y=\sin\left(x\right)e^x$y=sin(x)ex. Give your answer in factorised form.

Question 2

Find the derivative of $y=\cos\left(\ln x\right)$y=cos(lnx).

Question 3

Find the derivative of $\ln\left(\sin x\right)$ln(sinx).

 

Outcomes

M8-11

Choose and apply a variety of differentiation, integration, and antidifferentiation techniques to functions and relations, using both analytical and numerical methods

91578

Apply differentiation methods in solving problems

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