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2.07 Further differentiation

Lesson

We have seen many cases where a combination of rules can be applied to find a derivative. Let's now expand on the types of problems encountered and use a combination of rules across each of the sections covered thus far. To find a derivative for more complex functions, identify the types of functions in the problem - powers, polynomials, exponential and trigonometric, and then look for structure in the function that will dictate what rules need to be applied - such as product, quotient or composites of functions.

Below is a reference list of the rules we have learned so far:

Rules of differentiation
  • Product rule: If $y=uv$y=uv, then $y'=uv'+vu'$y=uv+vu
  • Quotient rule: If $y=\frac{u}{v}$y=uv, then $y'=\frac{vu'-uv'}{v^2}$y=vuuvv2
  • Chain rule: If a function is given by $y=f\left(u\right)$y=f(u), where $u=g\left(x\right)$u=g(x) then $\frac{dy}{dx}=\frac{dy}{du}\times\frac{du}{dx}$dydx=dydu×dudx
  • Function to a power: (Special case of the chain rule) If $y=\left[f\left(x\right)\right]^n$y=[f(x)]n then $\frac{dy}{dx}=nf'\left(x\right)\ \left[f\left(x\right)\right]^{n-1}$dydx=nf(x) [f(x)]n1

And also a summary of derivatives of some special functions and properties of derivatives:

Summary
Function Derivative
$kf\left(x\right)$kf(x), where $k$k is a constant $kf'\left(x\right)$kf(x)
$f\left(x\right)+g\left(x\right)$f(x)+g(x) $f'\left(x\right)+g'\left(x\right)$f(x)+g(x)
$x^n$xn $nx^{n-1}$nxn1
$e^x$ex $e^x$ex
$e^{f\left(x\right)}$ef(x) $f'\left(x\right)e^{f\left(x\right)}$f(x)ef(x)
$\sin\left(x\right)$sin(x) $\cos\left(x\right)$cos(x)
$\sin\left(f\left(x\right)\right)$sin(f(x)) $f'\left(x\right)\cos\left(f\left(x\right)\right)$f(x)cos(f(x))
$\cos\left(x\right)$cos(x) $-\sin\left(x\right)$sin(x)
$\cos\left(f\left(x\right)\right)$cos(f(x)) $-f'\left(x\right)\sin\left(f\left(x\right)\right)$f(x)sin(f(x))

Worked example

Differentiate $y=e^{\sin x}+x^2\cos x$y=esinx+x2cosx.

Think: We have a range of exponential, power and trigonometric functions. The first term is a composite function of a trigonometric function within an exponential function and hence we will need to apply the chain rule. The second term is a product of a power function with a trigonometric function and hence we will need to apply the product rule.

Do:

  • First term is of the form $e^{f\left(x\right)}$ef(x) so its derivative will be of the form $f'\left(x\right)e^{f\left(x\right)}$f(x)ef(x), with:
$f\left(x\right)$f(x) $f'\left(x\right)$f(x)
$\sin x$sinx $\cos x$cosx
  • Second term is of the form $uv$uv and so it derivative will be of the form $uv'+vu'$uv+vu:
Let $u=x^2$u=x2 then $u'=2x$u=2x
and $v=\cos x$v=cosx then $v'=-\sin x$v=sinx

Hence,

$\frac{dy}{dx}$dydx $=$= $f'\left(x\right)e^{f\left(x\right)}+uv'+vu'$f(x)ef(x)+uv+vu  
  $=$= $\cos xe^{\sin x}+x^2\left(-\sin x\right)+\left(\cos x\right)\left(2x\right)$cosxesinx+x2(sinx)+(cosx)(2x)

Make appropriate substitutions

  $=$= $\cos xe^{\sin x}-x^2\sin x+2x\cos x$cosxesinxx2sinx+2xcosx  

 

 

Practice questions

Question 1

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

Question 2

Consider the expression $\frac{4x^2+e^x}{\cos7x}$4x2+excos7x.

  1. By letting $u=4x^2+e^x$u=4x2+ex, find $u'$u.

  2. By letting $v=\cos\left(7x\right)$v=cos(7x), find $v'$v.

  3. Hence, find the derivative of $\frac{4x^2+e^x}{\cos\left(7x\right)}$4x2+excos(7x).

Question 3

Find the equation of the tangent to the curve $y=e^{\cos x}$y=ecosx at the point $x=\frac{3\pi}{2}$x=3π2.

 

Outcomes

ACMMM105

understand the notion of composition of functions and use the chain rule for determining the derivatives of composite functions

ACMMM106

apply the product, quotient and chain rule to differentiate functions such as xe^x, tan⁡x,1/x^n, x sin⁡x, e^(−x)sin⁡x and f(ax+b)

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