6H - More on differentiation by rule
Using the chain rule to generalise differentiation:
The chain rule can be used to generalise rule for differentiation for our elementary functions:
- If \(f(x)=e^{g(x)}\), then \(f'(x)=g'(x)e^{g(x)}\).
- If \(f(x)=log_e(g(x))\), then \(f'(x)=\frac{g'(x)}{g(x)}\).
- If \(f(x)=sin(g(x))\), then \(f'(x)=g'(x)cos(g(x))\).
- If \(f(x)=cos(g(x))\), then \(f'(x)=-g'(x)sin(g(x))\).
- If \(f(x)=tan(g(x))\), then \(g'(x)sec^2(g(x))\).
6H - Example 1: Generalising differentiation using the chain rule
Determine the derivative of each of the following:
(a) \(f(x)=e^{cos(x)}\) (b) \(g(x)=log_e(x^2-1)\) (c) \(h(x)=cos(\sqrt{x})\) (d) \(y=tan(2x^2-3x)\) |
6H - Example 1: Video solution
6H - Example 1: Practice
Question 1: ABC Question 2: ABC 6H - Example 1: Solutions
Question 1: ABC Question 2: ABC |
Combining the chain product and/or quotient rules:
In some cases, more than one of the chain, product or quotient rules may need to be applied within a question.
6H - Example 2: Combining the chain, product and quotient rules
If \(f(x)=\sqrt{sin(x)cos(x)}\), find \(f'(x)\).
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6H - Example 2: Video solution
6H - Example 2: Practice
Question 1: ABC Question 2: ABC 6H - Example 2: Solutions
Question 1: ABC Question 2: ABC |
6H - Example 3: Combining the chain, product and quotient rules
Calculate the derivative, \frac{dy}{dx}, for the function \[y=\frac{2x^2-5x}{\sqrt{x^2+2}}\]
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6H - Example 3: Video solution
6H - Example 3: Practice
Question 1: ABC Question 2: ABC 6H - Example 3: Solutions
Question 1: ABC Question 2: ABC |