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Differentiation of exponentials and logarithms for SQA Higher Maths

Differentiation of exponentials and logarithms

This page covers the following topics:

1. Differentiating basic exponents
2. Differentiating exponents
3. Differentiating natural logarithm
4. Differentiating logarithms

The function f(x) = eหฃ is identical to its derivative; use the chain rule to differentiate variables in the form eยชหฃ โบ แต‡. The first derivative is used to find the gradient at a given value of x.

Differentiating basic exponents

To differentiate variables in the form aหฃ, use the derivative exponent rule and the chain rule. To find the gradient of an exponential graph, use the first derivative.

Differentiating exponents

To differentiate equations and variables in the form lnx, use the following rule. To find the gradient of a natural logarithmic graph, use the first derivative.

Differentiating natural logarithm

The derivative of logarithmic functions can only be found when the base, b, is a positive real number and b โ‰  1. The first derivative is used to find the gradient, m, at any point of the logarithmic curve.

Differentiating logarithms

1

Evaluate (d/dx)3logโ‚…(4x).

3/xln(5)

Evaluate (d/dx)3logโ‚…(4x).

2

Find the first derivative of the function f(x) = ln(3x โˆ’ 5).

3/(3x โˆ’ 5)

Find the first derivative of the function f(x) = ln(3x โˆ’ 5).

3

Find the gradient of the curve y = logโ‚โ‚€(3x + 1) when x = 0.25.

(d/dx)logโ‚โ‚€(3x + 1) = 3/(3x + 1)ln(10). Substitute x = 0.25. The answer is 0.740 (to 3 s.f.).

Find the gradient of the curve y = logโ‚โ‚€(3x + 1) when x = 0.25.

4

Find the gradient of the graph y = lnโˆš(5 โˆ’ x) at the point x = 2.

(d/dx) (ln(5 โˆ’ x)) = (d/dx) (0.5ln(5 โˆ’ x)) = โˆ’ (1/2(5 โˆ’ x)). When x = 2, the gradient = โˆ’3.

Find the gradient of the graph y = lnโˆš(5 โˆ’ x) at the point x = 2.

5

Differentiate y = lnx/x with respect to x.

(1 โˆ’ lnx)/xยฒ

Differentiate y = lnx/x with respect to x.

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