The function f(x) = eˣ is identical to its derivative; use the chain rule to differentiate variables in the form eªˣ ⁺ ᵇ. The first derivative is used to find the gradient at a given value of x.

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

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.

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

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3/xln(5)

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3/(3x − 5)

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(d/dx)log₁₀(3x + 1) = 3/(3x + 1)ln(10). Substitute x = 0.25. The answer is 0.740 (to 3 s.f.).

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

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(1 − lnx)/x²