First principles: x⁴ and higher power terms

Questions asking students to differentiate from first principles where the function is of the form xⁿ for n≥4 (e.g., x⁴)

4 questions · Moderate -0.3

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Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
1. Expand \((2 + h)^4\). [3]
2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
3. Show how your result in part (iii) \((B)\) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]

Find the equation of the tangent to the curve \(y = x^4\) at the point where \(x = 2\). Give your answer in the form \(y = mx + c\). [4]
Calculate the gradient of the chord joining the points on the curve \(y = x^4\) where \(x = 2\) and \(x = 2.1\). [2]
1. Expand \((2 + h)^4\). [3]
2. Simplify \(\frac{(2 + h)^4 - 2^4}{h}\). [2]
3. Show how your result in part (iii) (B) can be used to find the gradient of \(y = x^4\) at the point where \(x = 2\). [2]

Expand \((x + h)^4\) [2 marks]
Hence, find an expression, in terms of \(x\) and \(h\), for the gradient of the line \(PQ\) [1 mark]
Explain how to use the answer from part (b) to obtain the gradient function of \(y = x^4\) [2 marks]