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Derivatives

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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