11. Differentiation

iGCSE (2021 Edition)

Lesson

As we have seen with other functions, we can use our knowledge of differentiating logarithmic functions to find the equation of tangents and normals to such functions.

Differentiating $y=\ln f\left(x\right)$`y`=`l``n``f`(`x`)

$\frac{d}{dx}\ln f\left(x\right)$ddxlnf(x) |
$=$= | $\frac{f'\left(x\right)}{f\left(x\right)}$f′(x)f(x) |

When finding equations of tangents and normals, remember that the gradient of the tangent at any point on a function is given by the first derivative of the function. The following formulas are also useful:

Formulas for tangents and normals

Given the gradient $m$`m` and a point $\left(x_1,y_1\right)$(`x`1,`y`1), the equation of a line can be found using the point gradient formula:

$y-y_1=m\left(x-x_1\right)$`y`−`y`1=`m`(`x`−`x`1)

Given the gradient of the tangent $m_1$`m`1 at a point, the gradient of the normal $m_2$`m`2 is:

$m_2=-\frac{1}{m_1}$`m`2=−1`m`1

Find the gradient of the tangent to the curve $y=\ln\left(4x-8\right)$`y`=`l``n`(4`x`−8) at the point where $x=3$`x`=3.

**Solution:** Finding the derivative of $y=\ln\left(4x-8\right)$`y`=`l``n`(4`x`−8) gives:

$\frac{dy}{dx}$dydx |
$=$= | $\frac{4}{4x-8}$44x−8 |

$=$= | $\frac{1}{x-2}$1x−2 |

So at $x=3$`x`=3, the gradient of the tangent is:

$\frac{dy}{dx}$dydx |
$=$= | $\frac{1}{3-2}$13−2 |

$=$= | $1$1 |

Consider the function $f\left(x\right)=\ln x$`f`(`x`)=`l``n``x`.

Determine the $x$

`x`value of the $x$`x`-intercept.Determine the equation of the tangent line to the curve at the point where it crosses the $x$

`x`-axis.Determine the equation of the normal line to the curve at the point where it crosses the $x$

`x`-axis.

The first and second derivatives of logarithmic functions can be used to determine concavity, and regions where a logarithmic function is increasing or decreasing. Additionally, any stationary points and points of inflection can be found. Using these tools we can sketch functions involving logarithms.

Consider the graph of $y=\ln x$`y`=`l``n``x`.

Loading Graph...

Is the function increasing or decreasing?

Increasing

ADecreasing

BIncreasing

ADecreasing

BIs the gradient to the curve negative at any point on the curve?

No

AYes

BNo

AYes

BWhich of the following best completes this sentence?

"As $x$

`x`increases, the gradient of the tangent..."decreases at a constant rate.

Aincreases at an increasing rate.

Bincreases at a constant rate.

Cdecreases at an increasing rate.

Dincreases at a decreasing rate.

Edecreases at a decreasing rate.

Fdecreases at a constant rate.

Aincreases at an increasing rate.

Bincreases at a constant rate.

Cdecreases at an increasing rate.

Dincreases at a decreasing rate.

Edecreases at a decreasing rate.

FWhich of the following best completes the sentence?

"As $x$

`x`gets closer and closer to $0$0, the gradient of the tangent..."increases towards a fixed value.

Adecreases towards $-\infty$−∞.

Bdecreases towards $0$0.

Cincreases towards $\infty$∞.

Dincreases towards a fixed value.

Adecreases towards $-\infty$−∞.

Bdecreases towards $0$0.

Cincreases towards $\infty$∞.

DWe have found that the gradient function must be a strictly positive function, and it must also be a function that decreases at a decreasing rate. What kind of function could it be?

Quadratic, of the form $y'=ax^2$

`y`′=`a``x`2.AExponential, of the form $y'=a^{-x}$

`y`′=`a`−`x`.BLinear, of the form $y=ax$

`y`=`a``x`.CHyperbolic, of the form $y'=\frac{a}{x}$

`y`′=`a``x`.DQuadratic, of the form $y'=ax^2$

`y`′=`a``x`2.AExponential, of the form $y'=a^{-x}$

`y`′=`a`−`x`.BLinear, of the form $y=ax$

`y`=`a``x`.CHyperbolic, of the form $y'=\frac{a}{x}$

`y`′=`a``x`.D

Consider the function $f\left(x\right)=x-\ln x$`f`(`x`)=`x`−`l``n``x`.

Find the values of $x$

`x`where $f'\left(x\right)=0$`f`′(`x`)=0.Find $f''\left(x\right)$

`f`′′(`x`).Determine the nature of the stationary point.

Minimum

APoint of inflection

BMaximum

CMinimum

APoint of inflection

BMaximum

CWhich of the following best describes the behaviour of this function?

As $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$-\infty$−∞.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$-\infty$−∞.AAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$0$0.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$\infty$∞.BAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$\infty$∞.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$0$0.CAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$\infty$∞.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$\infty$∞.DAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$-\infty$−∞.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$-\infty$−∞.AAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$0$0.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$\infty$∞.BAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$\infty$∞.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$0$0.CAs $x$

`x`$\to$→$\infty$∞, $y$`y`$\to$→$\infty$∞.As $x$

`x`$\to$→$0$0, $y$`y`$\to$→$\infty$∞.DWhich is the correct sketch of the graph?

Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...DLoading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D

Use the derivatives of the standard functions sinx, cos x, tan x, together with constant multiples, sums and composite functions.

Use the derivatives of the standard functions e^x, ln x, together with constant multiples, sums and composite functions.