1 of 8 0-day streak

Differentiate natural-logarithmic functions

Question 1 of 8

No calculator

Consider the function with the displayed rule

f(x)=-3\,\log_e\left(-3\,x + 2\right)

The target x-value is shown below

x=\frac{1}{3}

(a)1 mark

Use the natural logarithm derivative rule to obtain the derivative

f'(x)=\frac{-3\,\left(-3\,x + 2\right)}{-3}

f'(x)=\frac{-3}{-3\,x + 2}

f'(x)=\frac{9}{-3\,x + 2}

f'(x)=\frac{-3}{\left(-3\right)\left(-3\,x + 2\right)}

(b)2 marks

Enter the signed numerical values of the labelled constants. Give exact values

h(x)=-3\,x + 2

A is the constant multiplying the original logarithm

B is the leading coefficient of the inner derivative, written in descending powers of x

f'(x)=\frac{A\,h'(x)}{h(x)}

\(A\): Original logarithm multiplier \(B\): Inner derivative coefficient

Use the derivative to find the gradient at the displayed point

x=\frac{1}{3}

Exact gradient

Format: Give an exact value