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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

3.2.4.1

select and apply the product rule, quotient rule and chain rule to differentiate functions; express derivatives in simplest and factorised form

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