Lesson

Derivatives

The derivative $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ measures the rate of change of $y$ with respect to $x$. Use the power rule, the chain rule, the product rule, and the quotient rule as appropriate.

Quiz

Q1

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ for $y = 6x^3 - \ln(x)$

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Differentiate each term; $\frac{d}{dx}\ln x = 1/x$.

Q2

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ for $y = \dfrac{1}{x^3}$

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Rewrite as $x^{-3}$ and apply the power rule.

Q3

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ for $y = e^{2x}$

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Chain rule: multiply by the derivative of the exponent.

Q4

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ for $y = (x^7 - 3x)^4$

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Chain rule with the power rule.

Q5

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ for $y = 7x^3 \sin(x)$

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Product rule.

Q6

Find $\dfrac{\mathrm{d}y}{\mathrm{d}x}$ for $y = \dfrac{\sqrt{x}}{\cos(x)}$

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Quotient rule; $\frac{d}{dx}\sqrt{x} = \frac{1}{2\sqrt{x}}$.