Differential Calculus

Differential calculus extends the Year 11 rules (product, quotient and chain rules for polynomials and powers) to exponential, logarithmic and trig functions. On its own it was worth 7 marks on NESA's sample paper for the 2024 syllabus, but almost every calculus question in Applications of calculus and Integral calculus depends on it, so speed and accuracy here pay off everywhere.

• Differentiate $k{e}^{ax}$, $e^{ax+b}$, $a^x$ and $e^{f(x)}$, and prove the rules for $e^{ax+b}$ and $a^x$.
• Differentiate $\ln x$, ${\log }_{a}x$ and $\ln f(x)$, and prove the rules for $\ln x$ and ${\log }_{a}x$.
• Differentiate $\sin x$, $\cos x$, $\tan x$, $\sec x$, $\mathrm{cosec}x$, $\cot x$ and the chain-rule versions $\sin f(x)$, $\cos f(x)$, $\tan f(x)$.
• Combine the product, quotient and chain rules with any function in the course, and use derivatives to solve problems (gradients, tangents and normals, rates).

All trig calculus uses radians. The derivative of $\sin x$ is $\cos x$ only when $x$ is in radians.

Learn these until they are automatic. A reference sheet is provided in the exam, but looking results up costs time.

Chain rule versions

If $u=f(x)$, every result above becomes "derivative of the outside, times $f^{\prime }(x)$":

The product and quotient rules

Write down $u$, $v$, $u^{\prime }$ and $v^{\prime }$ before you substitute. It keeps the working tidy and earns marks even if the final simplification slips.

Proof for $e^{ax+b}$. Let $u=ax+b$, so $\frac{du}{dx}=a$. By the chain rule, $\frac{d}{dx}{e}^u=e^u\frac{du}{dx}=a{e}^{ax+b}$.

Proof for the exponential rule. To differentiate $a^x$, note that since $a=e^{\ln a}$, write $a^x=e^{x\ln a}$. The exponent $x\ln a$ is linear with gradient $\ln a$, so the chain rule gives $\frac{d}{dx}{a}^x=(\ln a)e^{x\ln a}=a^x\ln a$.

Proof for the natural log rule. Differentiate both sides of $e^{\ln x}=x$ using the chain rule: $e^{\ln x}\cdot \frac{d}{dx}\ln x=1$, so $\frac{d}{dx}\ln x=\frac{1}{e^{\ln x}}=\frac{1}{x}$ for $x>0$.

Proof for logs to base a. To differentiate ${\log }_{a}x$, use the change of base law, ${\log }_{a}x=\frac{\ln x}{\ln a}$. Since $\ln a$ is a constant,

Before differentiating a logarithm, use log laws to simplify. For example $\ln \left(x^3\sqrt{x+1}\right)=3\ln x+\frac{1}{2}\ln (x+1)$, which differentiates without the product or chain rule tangle.

Worked example 1

Differentiate each function.

(a) $y=\ln (x^3-2x)$. With $f(x)=x^3-2x$:

(b) $y=4e^{-x/2}$. The exponent has derivative $-\frac{1}{2}$:

(c) $y={\log }_{5}x$:

(d) Find the gradient of $y=2^x$ at $x=3$. Here $y^{\prime }=2^x\ln 2$, so the gradient is $8\ln 2\approx 5.545$.

The syllabus wants you to see informally, from the graphs, why $\frac{d}{dx}\sin x=\cos x$: the gradient of $y=\sin x$ is 1 at $x=0$, 0 at $x=\frac{\pi }{2}$ and $-1$ at $x=\pi$, which traces out $y=\cos x$. Similarly the gradient of $\cos x$ traces out $-\sin x$.

The others follow from the quotient rule. For $\tan x=\frac{\sin x}{\cos x}$:

For the reciprocal functions, write $\mathrm{cosec}x=(\sin x{)}^{-1}$ and use the chain rule:

The same method gives the $\sec x$ and $\cot x$ results. A useful memory aid: every "co" function ($\cos$, $\mathrm{cosec}$, $\cot$) has a negative derivative.

Watch powers of trig functions: ${\sin }^{2}x$ means $(\sin x{)}^2$, so its derivative needs the chain rule, $2\sin x\cos x$.

Worked example 2

Differentiate $y=x^2\cos 3x$.

Let $u=x^2$ and $v=\cos 3x$, so $u^{\prime }=2x$ and $v^{\prime }=-3\sin 3x$.

Worked example 3

(a) Differentiate $y=\frac{e^{2x}}{x+1}$.

Let $u=e^{2x}$ and $v=x+1$, so $u^{\prime }=2e^{2x}$ and $v^{\prime }=1$.

(b) Find where the tangent to this curve is horizontal.

$\frac{dy}{dx}=0$ when $e^{2x}(2x+1)=0$. Since $e^{2x}>0$ for all $x$, we need $2x+1=0$, so $x=-\frac{1}{2}$. Then $y=\frac{e^{-1}}{1/2}=\frac{2}{e}$, giving the point $\left(-\frac{1}{2},\frac{2}{e}\right)$.

Factorising out the exponential is the key step. It is never zero, which is why it can be divided out.

Typical problems in this focus area:

• Find the equation of a tangent or normal at a point: get the gradient from $f^{\prime }(a)$, then use $y-y_1=m(x-x_1)$. A normal has gradient $-\frac{1}{m}$.
• Find where a curve has a given gradient, or where two curves have parallel tangents.
• Show that a function satisfies an equation involving its derivatives, such as $y^{\prime \prime }+4y=0$ for $y=\sin 2x$.
• Find a rate of change at a given time (more of this in MA12-AC).

• Using degrees. Calculus results for trig functions are only true in radians.
• Writing $\frac{d}{dx}{e}^{3x}=3x{e}^{3x}$ or $e^{3x}$: the derivative of the exponent multiplies, it does not get inserted.
• Treating $\ln (x^2+1)$ like $\ln x$ and writing $\frac{1}{x^2+1}$ without the chain-rule factor $2x$.
• Swapping the order in the quotient rule numerator ($u{v}^{\prime }-u^{\prime }v$ has the wrong sign).
• Writing $\frac{d}{dx}\cos x=\sin x$ (it is $-\sin x$).
• Confusing $a^x$ (exponential, derivative $a^x\ln a$) with $x^n$ (power, derivative $n{x}^{n-1}$).
• Not using log laws first, then making an algebra error inside a long quotient rule.

• Differentiate: usually 1–2 marks. One mark is often for correct use of the rule (product, quotient or chain), the second for the correct simplified answer.
• Show that / prove: start from a definition or identity the syllabus names ($e^{\ln x}=x$, $a=e^{\ln a}$, change of base), and show every step until you reach exactly the given result.
• Leave answers in a factorised, simplified form, especially if the next part asks you to solve $f^{\prime }(x)=0$.
• Give exact answers (like $8\ln 2$ or $\frac{2}{e}$) unless the question asks for a decimal.
• Check the domain of a log function: $\ln f(x)$ is only defined where $f(x)>0$.

• MA11-DI Introduction to differentiation: first principles, the product, quotient and chain rules, and tangents and normals for polynomials come from here.
• MA11-EL Exponential and logarithmic functions: log laws and the change of base formula are used in proofs and simplification.
• MA12-AC Applications of calculus: stationary points, concavity, optimisation, motion and exponential growth all start with these derivatives.
• MA12-IC Integral calculus: each derivative here, read backwards, is an integral.