7.3 Differentiating powers, exponentials, trigonometric functions and logarithms — Practise

Work through these on your own. Rewrite every root and fraction as a power before you differentiate. Keep the constants that come from inside a function, and watch the minus sign on cosine.

Both the questions and the answers below come from a set that is checked by computer every time the book is built.

Differentiate $y = 8x^3$ .
Differentiate $y = \dfrac{5}{x}$ .
Differentiate $y = x^{1/4}$ .
Differentiate $y = 2x^5 - 4x^3 + x$ .
Differentiate $y = 3x^{-2}$ .
Differentiate $y = e^{4x}$ .
Differentiate $y = 3e^{-2x}$ .
Differentiate $y = 2^{3x}$ .
Differentiate $y = \sin 5x$ .
Differentiate $y = 4\cos 2x$ .
Differentiate $y = \tan 3x$ .
Differentiate $y = 3\ln x + 2e^x - \cos x$ .

Answers and concise solutions

1. The derivative

Answer: Multiply by the power $3$ and reduce it by one: $\dfrac{dy}{dx} = 8 \times 3x^2 = 24x^2$ .

2. The derivative

Answer: Write $y = 5x^{-1}$ . Then $\dfrac{dy}{dx} = -5x^{-2} = -\dfrac{5}{x^2}$ . The derivative is negative everywhere, so the curve falls on each branch.

3. The derivative

Answer: Apply the power rule: $\dfrac{dy}{dx} = \tfrac{1}{4}x^{-3/4}$ . The new power is negative, so the derivative can also be written $\dfrac{1}{4x^{3/4}}$ .

4. The derivative

Answer: Differentiate term by term: $10x^4 - 12x^2 + 1$ . The derivative of $x$ is $1$ , because $x = x^1$ .

5. The derivative

Answer: Multiply by $-2$ and reduce the power by one: $\dfrac{dy}{dx} = -6x^{-3} = -\dfrac{6}{x^3}$ . Each branch of the curve falls.

6. The derivative

Answer: The derivative of $e^{kx}$ is $ke^{kx}$ , so $\dfrac{dy}{dx} = 4e^{4x}$ . The constant $4$ appears in front of the same exponential.

7. The derivative

Answer: The constant in the exponent is $-2$ , so the derivative is $3 \times (-2)e^{-2x} = -6e^{-2x}$ . The curve falls, which matches the negative derivative.

8. The derivative

Answer: For $a^{kx}$ the derivative is $ka^{kx}\ln a$ . Here $a = 2$ and $k = 3$ , so $\dfrac{dy}{dx} = 3 \times 2^{3x}\ln 2$ .

9. The derivative

Answer: The derivative of $\sin kx$ is $k\cos kx$ , so $\dfrac{dy}{dx} = 5\cos 5x$ .

10. The derivative

Answer: The derivative of $\cos kx$ is $-k\sin kx$ . Here $k = 2$ , so $\dfrac{dy}{dx} = 4 \times (-2\sin 2x) = -8\sin 2x$ .

11. The derivative

Answer: The derivative of $\tan kx$ is $k\sec^2 kx$ . Here $k = 3$ , so $\dfrac{dy}{dx} = 3\sec^2 3x = \dfrac{3}{\cos^2 3x}$ .

12. The derivative

Answer: Differentiate term by term: $\dfrac{3}{x} + 2e^x + \sin x$ . The minus sign in front of $\cos x$ becomes a plus, because the derivative of $\cos x$ is $-\sin x$ .