This tutorial covers material encountered in chapter 9 of the VCE Mathematical Methods Textbook, namely:
- The derivative of functions seen previously in tutorial worksheets 1-5
- Chain, Product and Quotient rules applied to the aforementioned functions
Q1 – Derivatives of Polynomial Functions
Q2 – Gradients of Tangents of Polynomial Functions
Q3 – Derivatives of Exponential, Logarithmic, Sine & Cosine Functions
Q4 – Gradients of Tangents of Exponential, Logarithmic, Sine & Cosine Functions
Q5 – Finding Derivative Zeros
Worksheet
Q1. Find the derivatives of the following with respect to \(x\):
(a) \(f(x)=x+\sqrt{2-x^2}\)
(b) \(g(x)=\dfrac{x^2-3x+4}{2x^2+1}\)
(c) \(h(x)=(x+3)\sqrt{4x+3}\)
(d) \(j(x)=\sqrt[3]{5x^2-7}\)
(e) \(k(x)=(5x^2-7)^{\frac{1}{3}}\)
Q2. Find the gradient of the tangent of each function at the corresponding points:
(a) \(a(x)=4x^2-4x+1,\) at \(x=1\)
(b) \(b(x)=\dfrac{x-3}{x^4+2},\) at \(x=-1\)
(c) \(c(x)=\left(2x^2+3 \right)^{\frac{2}{3}}\) at \(x=4\)
Q3. Find the derivatives of the following with respect to \(x\):
(a) \( f(x)=\ln(x+3)+3 \)
(b) \(g(x)=\sin(2x)\)
(c) \(h(x)=x^2\cos(3x)\)
(d) \(j(x)=\dfrac{\sin(2x+1)}{\cos(2x+1)}\)
(e) \(k(x)=e^{2x}\sin(3x)\)
Q4. Find the gradient of the tangent of each function at the corresponding points:
(a) \(y(x)=\cos(\pi x)\) at \(x=\dfrac{1}{6}\)
(b) \(t(x)=xe^{2x},\) at \(x=3\)
(c) \(r(x)=(4x^2-2)\ln(x-2),\) at \(x=3\)
(d) \(w(x)= -x\sin(3x),\) at \(x=\dfrac{\pi}{4}\)
Q5. For what values of \(x\in\R,\) is the derivative of \(f(x)=\left(2x+\dfrac{3}{x}\right)^2\) with respect to \(x\) zero?
Q6. If the function \(f\) is differentiable for all real numbers, find the derivative of each of the following:
(i) \(xf(x)\)
(ii) \(\dfrac{1}{f(x)}\)
(iii) \(\dfrac{x}{f(x)}\)
(iv) \(\dfrac{x^2}{\left[f(x)\right]^2}\)
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