IGCSE Add Maths Derivative of a Natural Logarithm Part 1 2025J

​Derivative of a Natural Logarithm

​If x > 0 and y = ln x, then

​Derivative of a Natural Logarithm

​If x > 0 and y = ln 3x, then

​Derivative of a Natural Logarithm

​If x > 0 and y = ln (3x+2), then

​Differentiate the function with respect to x

Multiple Choice

Differentiate the function with respect to x.

y = ln 5x

• $$\frac{1}{5x}\times5$$  

• 5x

• 5

• $$\frac{5}{x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = ln (-2x)

• $$\frac{-1}{2x}\times-2$$  

• $$\frac{-x}{2}$$  

• $$\frac{1}{2x}$$  

• $$\frac{-2}{x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = ln 7x

• $$\frac{1}{7x}\times7$$  

• $$\frac{x}{7}$$  

• $$\frac{1}{-7x}$$  

• $$\frac{-7}{x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = ln (x +1)

• $$\frac{1}{x+1}$$  

• $$\frac{x+1}{x}$$  

• $$\frac{1}{1-x}$$  

• $$\frac{1}{x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = ln (7x +1)

• $$\frac{1}{7x+1}\ \times7\ =\frac{7}{7x+1}$$  

• $$\frac{7x+1}{x}$$  

• $$\frac{1}{1-7x}$$  

• $$\frac{1}{7x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln3x^2$$  

• $$\frac{1}{3x^2}\times6x\ =\ \frac{2}{x}$$  

• $$\frac{1}{3x^2}$$  

• $$\frac{1}{-7x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln6x^4$$  

• $$\frac{1}{6x^4}\times24x^3\ =\frac{4}{x}$$  

• $$\frac{1}{6x^4}\times24x\ =\frac{4}{x^3}$$  

•   $$\frac{1}{6x^4}\times24x^5\ =\frac{4x}{1}$$

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(3x-2\right)^2$$  

(

(tips: diff' the ln, then index, then bracket)

• $$\frac{1}{\left(3x-2\right)^2}\ \times2\left(3x-2\right)\left(3\right)$$  

•   $$\frac{1}{\left(3x-2\right)^2}\ \times\left(3x-2\right)\left(3\right)$$

• $$\frac{1}{\left(3x-2\right)^2}\ \times2\left(3x-2\right)$$

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(4x-3\right)^2$$  

(tips: diff' the ln, then index, then bracket)

• $$\frac{1}{\left(4x-3\right)^2}\ \times2\left(4x-3\right)\left(4\right)$$  

•   $$\frac{1}{\left(4x-3\right)^2}\ \times\left(4x-3\right)\left(4\right)$$

• $$\frac{1}{\left(4x-3\right)^2}\ \times2\left(4x-3\right)$$

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(5+4x\right)^3$$  

(tips: diff' the ln, then index, then bracket)

•   $$\frac{1}{\left(5+4x\right)^3}\ \times3\left(5+4x\right)\left(2\right)$$

• $$\frac{1}{\left(5+4x\right)^3}\ \times3\left(5+4x\right)\left(4\right)$$

• $$\frac{1}{\left(5+4x\right)^3}\ \times3\left(5+4x\right)^2\left(4\right)$$

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(8-x^3\right)$$  

(tips: diff' the ln, then index, then bracket)

• $$\frac{1}{\left(8-x^3\right)}\ \times\left(-3x^2\right)$$  

•   $$\frac{1}{8-x^3}\ \times\left(8-x^3\right)\left(-1\right)$$

Multiple Choice

Differentiate the function with respect to x.

y = sin x

• sin x

• cos x

Multiple Choice

Differentiate the function with respect to x.

y = cos x

• sin x

• cos x

• - sin x

Multiple Choice

Differentiate the function with respect to x.

y = tan x

• $$\sec^2x$$  

• $$\cos^2x$$  

• $$\sin^2x$$  

Multiple Choice

Differentiate the function with respect to x.

y = cos 2x

(tips: differentiate trigo, then $$2x^{ }$$  )

• $$\sin2x\times\left(2\right)=2\sin2x$$  

• $$-\cos2x\times\left(2\right)=\ -2\cos2x$$  

• $$-\sin2x\times\left(2\right)\ =\ -2\sin2x$$  

Multiple Choice

Differentiate the function with respect to x.

y = sin 5x

(tips: differentiate trigo, then $$5x^{ }$$  )

• $$\sin5x\times\left(5\right)=5\sin5x$$  

• $$\cos5x\times\left(5\right)=\ 5\cos5x$$  

• $$-\sin5x\times\left(5\right)\ =\ -5\sin5x$$  

Multiple Choice

Differentiate the function with respect to x.

y = tan 5x

(tips: differentiate trigo, then $$5x^{ }$$  )

• $$\sin^25x\times\left(5\right)=5\sin^25x$$  

• $$\sec^25x\times\left(5\right)=\ 5\sec^25x$$  

• $$-\sin5x\times\left(5\right)\ =\ -5\sin5x$$  

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(\sin x\right)^{ }$$  

(tips: differentiate the ln, then trigo)

• $$\frac{1}{\sin x}\times\cos x$$  

• $$\frac{1}{\sin x}\times\sin x$$  

• $$\frac{1}{\sin x}\times\left(x\right)$$  

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(\cos x\right)^{ }$$  

(tips: differentiate the ln, then trigo, then $$x^{ }$$  )

• $$\frac{1}{\sin x}\times\cos x$$  

• $$\frac{1}{\cos x}\times\left(-\sin x\right)$$  

• $$\frac{1}{\sin x}\times\left(x\right)$$  

Multiple Choice

Differentiate the function with respect to x.

y = $$\ln\left(\cos x^2\right)$$  

(tips: differentiate the ln, then trigo, then $$x^2$$  )

•   $$\frac{1}{\cos x^2}\times\left(-\sin x^2\right)\times\left(2\right)=\frac{-2\sin x^2}{\cos x^2}$$

• $$\frac{1}{\cos x^2}\times\left(-\sin x^2\right)\times\left(2x\right)=\frac{-2x\sin x^2}{\cos x^2}$$

Multiple Choice

Differentiate the function with respect to x.

y = $$e^{2x}$$  

(tips: copy the exponential, then differentiate the index,)

•   $$\frac{\text{d}y}{\text{d}x}=\ e^{2x}\times\left(2\right)=2e^{2x}$$  

•   $$\frac{\text{d}y}{\text{d}x}=\ e^{2x}\times\left(2x\right)=2xe^{2x}$$  

Multiple Choice

Differentiate the function with respect to x.

y = $$e^{5x}$$  

(tips: copy the exponential, then differentiate the index,)

•   $$\frac{\text{d}y}{\text{d}x}=\ e^{5x}\times\left(5\right)=5e^{5x}$$  

•   $$\frac{\text{d}y}{\text{d}x}=\ e^{5x}\times\left(5x\right)=5xe^{5x}$$  

Multiple Choice

Differentiate the function with respect to x.

y =ln $$e^{5x}$$  

(tips: differentiate ln ,copy the exponential, then differentiate the index,)

•   $$\frac{\text{d}y}{\text{d}x}=\frac{1}{e^{5x}}\times\ e^{5x}\times\left(5\right)=\frac{5e^{5x}}{e^{5x}}=5$$  

•   $$\frac{\text{d}y}{\text{d}x}=\ e^{5x}\times\left(5x\right)=5xe^{5x}$$  

•   $$\frac{\text{d}y}{\text{d}x}=\frac{1}{e^{5x}}\times\ e^{5x}=\frac{e^{5x}}{e^{5x}}=1$$