SQA Advanced Higher Mathematics

## Exponential and natural logarithm functions The exponential function $e^x$ is its own derivative, and the natural logarithm is the function whose derivative is the reciprocal: $$ \frac{d}{dx}\left(e^x\right) = e^x, \qquad \frac{d}{dx}\left(\ln x\right) = \frac{1}{x} \quad (x > 0). $$ Combined with the chain rule these give the two results used most often. For any differentiable $f(x)$, $$ \frac{d}{dx}\left(e^{f(x)}\right) = f'(x)\,e^{f(x)}, \qquad \frac{d}{dx}\left(\ln f(x)\right) = \frac{f'(x)}{f(x)}. $$ So $\frac{d}{dx}\left(e^{3x^2 - x}\right) = (6x - 1)e^{3x^2 - x}$ and $\frac{d}{dx}\left(\ln(x^2 + 1)\right) = \frac{2x}{x^2 + 1}$. The second result is often easier after the logarithm laws have been used to split the expression. Since $\ln\left(\dfrac{x^3}{\sqrt{x+1}}\right) = 3\ln x - \tfrac{1}{2}\ln(x+1)$, its derivative is $\dfrac{3}{x} - \dfrac{1}{2(x+1)}$, with no quotient rule needed. ## The chain rule If $y = f(u)$ and $u = g(x)$, then $$ \frac{dy}{dx} = \frac{dy}{du}\times\frac{du}{dx}. $$ Differentiate the outer function with the inner function left alone, then multiply by the derivative of the inner function. Functions built from up to three layers need the rule up to twice in one line. For $y = \sin^3(2x)$ the layers are cube, sine and $2x$: $$ \frac{dy}{dx} = 3\sin^2(2x)\times\cos(2x)\times 2 = 6\sin^2(2x)\cos(2x). $$ For $y = e^{\sin(x^2)}$ the layers are exponential, sine and square: $$ \frac{dy}{dx} = e^{\sin(x^2)}\times\cos(x^2)\times 2x =…