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DIFFERENTIATION

Chain Rule | Product Rule | Quotient Rule | Sin Function | Cos Function | Tan Function | Exponential Function | Log Function |

ℹī¸ HSC Reference Sheet Differentiation Rules

At the back of every HSC Mathematics exam is a Reference Sheet containing a list of useful formulas for each topic. The rules for differentiation include:

\( \frac{d}{dx} \sin(f(x)) = f'(x) \cos(f(x)) \)

\( \frac{d}{dx} \cos(f(x)) = -f'(x) \sin(f(x)) \)

\( \frac{d}{dx} \tan(f(x)) = f'(x) \sec^{2}(f(x)) \)

\( \frac{d}{dx} e^{f(x)} = f'(x) e^{f(x)} \)

\( \frac{d}{dx} \log_{e}{f(x)} = \frac{f'(x)}{f(x)} \)

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🔄 Chain Rule Practice

\( y = f(x)^{n} \Rightarrow \frac{dy}{dx} = n \times f'(x) [f(x)]^{n} \)

Practice

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🔄 Product Rule Practice

\( y = uv \Rightarrow \frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx} \)

Practice

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🔄 Quotient Rule Practice

\( y = \frac{u}{v} \Rightarrow \frac{dy}{dx} = \frac{v \frac{du}{dx} - u \frac{dv}{dx}}{v^{2}} \)

Practice

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🔄 Sine Function Practice

\( y = \sin f(x) \Rightarrow \frac{dy}{dx} = f'(x) \cos f(x) \)

Example: \( \frac{d}{dx} \sin(3x^2) \)

Step 1: Recognise this function is in the form \( \sin(f(x)) \) with \( f(x) = 3x^2 \).

Step 2: The derivative of \( f(x) = 3x^2 \) is \( f'(x) = 6x \).

Step 3: Multiply by \( \cos(f(x)) \): \( y' = \cos(3x^2) \times 6x = \ 6x \cos(3x^2) \)

Practice

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🔄 Cosine Function Practice

\( y = \cos f(x) \Rightarrow \frac{dy}{dx} = - f'(x) \sin f(x) \)

Example: \( \frac{d}{dx} \cos(2x+1) \)

Step 1: Recognise this function is in the form \( \cos(f(x)) \) with \( f(x) = 2x+1 \).

Step 2: The derivative of \( f(x) = 2x+1 \) is \( f'(x) = 2 \).

Step 3: Multiply by \( \cos(f(x)) \): \( y' = -\sin(2x+1) \times 2 = -2 \sin(2x+1) \)

Practice

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🔄 Tan Function Practice

\( y = \tan f(x) \Rightarrow \frac{dy}{dx} = f'(x) \sec^{2} f(x) \)

Example: \( \frac{d}{dx} \tan(5x) \)

Step 1: Recognise this function is in the form \( \tan(f(x)) \) with \( f(x) = 5x \).

Step 2: The derivative of \( f(x) = 5x \) is \( f'(x) = 5 \).

Step 3: Multiply by \( \sec^{2}(f(x)) \): \( y' = \sec^{2}(5x) \times 5 = \ 5 \sec^{2}(5x) \)

Practice

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🔄 Exponential Function Practice

\( y = e^{f(x)} \Rightarrow \frac{dy}{dx} = f'(x)e^{f(x)} \)

\( y = a^{f(x)} \Rightarrow \frac{dy}{dx} = ln(a) \times f'(x)a^{f(x)} \)

Example: \( \frac{d}{dx} e^{x^{2}-3x-1} \)

Step 1: Recognise this function is in the form \( e^{f(x)} \) with \( f(x) = x^{2}-3x-1 \).

Step 2: The derivative of \( f(x) = x^{2}-3x-1 \) is \( f'(x) = 2x - 3 \).

Step 3: Multiply by \( e^{f(x)} \): \( y' = e^{x^{2}-3x-1} \times (2x - 3) = \ (2x - 3) e^{x^{2}-3x-1} \)

Practice

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🔄 Log Function Practice

\( y = \ln{f(x)} \Rightarrow \frac{dy}{dx} = \frac{f'(x)}{f(x)} \)

\( y = \log_{a}x \Rightarrow \frac{dy}{dx} = \frac{f'(x)}{( \ln a)f(x)} \)

Example: \( \frac{d}{dx} \ln(4 - x^{3}) \)

Step 1: Recognise this function is in the form \( ln(f(x)) \) with \( f(x) = 4 - x^{3} \).

Step 2: The derivative of \( f(x) = 4 - x^{3} \) is \( f'(x) = -3x^{2}\).

Step 3: Multiply by \( \frac{1}{f(x)} \): \( y' = -3x^{2} \times \frac{1}{4 - x^{3}} = \ -\frac{3x^{2}}{4 - x^{3}} \)

Practice

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"The only way to learn mathematics is to do mathematics" - Paul R Halmos