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Mathematics 1A Tutorial
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- Mathematics 1A (EN1911)
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- The Robert Gordon University (RGU)
Mathematics 1A Tutorial
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Differentiation Tutorial EN1911 (Mathematics 1A) 1Tutorial Questions on Differentiation
1. Differentiate each function from first principles, and evaluate the derivative for the
given value of x.
(a) 𝑓(𝑥) = 3𝑥 + 7 , find 𝑓’(1) (b) 𝑓(𝑥) = 𝑥 2 − 𝑥 , find 𝑓’(0)
1
(c) 𝑓(𝑥) = 𝑥 , find 𝑓’(2)
2. Find the gradients of the following curves at the given points
(a) 𝑃(1,3) on the curve 𝑓(𝑥) = 𝑥 2 + 2 (b) 𝑃(0, −1) on the curve 𝑓(𝑥) = 𝑥 3 − 1
2
(c) 𝑃(2,11) on the curve 𝑓(𝑥) = 2𝑥 + 3
3. Find the derivatives of the following functions:
1
(a) 𝑥 4 (b) 𝑥 −4 (c) 𝑥 2/3 (d) (e) 𝑥 7/2 (f) 𝑥 −1/4
𝑥2
3 1
(g) 𝑥 0.7 (h) √𝑥 (i)
√𝑥
4. Differentiate with respect to x:
𝑥4 7
(a) −14 (b) 10𝑥 2 (c) 5𝑥 3 (d) −2𝑥 −2 (e) (f)
4 𝑥3
7/4 6
(g) 6 √𝑥 (h) 8𝑥 (i) − √𝑥 3
5. Differentiate with respect to x:
(a) 𝑥 2 − 𝑥 3 (b) 4𝑥 2 − 3𝑥 (c) 2 + 5𝑥 − 7𝑥 2
1 1 1
(d) 3𝑥 3 − 5𝑥 2 + 𝑥 − 7 (e) 1 + 𝑥 + 𝑥 (f) 3 − 2𝑥 2 + 4𝑥 4
𝑑𝑦
6. Determine for each of the following:
𝑑𝑥
1
(a) 𝑦 = 3𝑥 2 + 5 (b) 𝑦 = 5𝑥 2 + 6𝑥 − 7 (c) 𝑦 = 𝑥 3 + 𝑥 3
4 1 5 5
(d) 𝑦 = 𝑥 − 𝑥 2 (e) 𝑦 = 5𝑥 3/2 − 𝑥1/2 (f) 𝑦 = √𝑥 − 5
𝑥 √
7. Differentiate with respect to the variable:
1+𝑤+𝑤 2
(a) 𝑥 2 (3 − 2𝑥) (b) (2𝑢 + 1)(𝑢 − 1) (c) 𝑤3
(2𝑣+1)(𝑣−1) 2𝑥−3 2𝑥 4 −3𝑥 2 −5
(d) (e) (f)
𝑣 √𝑥 𝑥3
8. Determine the second order derivative of each function:
2 7
(a) 2 − 3𝑥 + 5𝑥 2 (b) 5𝑥 3 − 6𝑥 2 + 2𝑥 − 6 (c) 3 − 𝑠 + 𝑠2
1+𝑤+𝑤 2
(d) 𝑢2 (3 − 2𝑢) (e) (2 + 𝑥)(3 + 5𝑥) (f) 𝑤3
9. Differentiate with respect to 𝑥:
(a) 3.14 sin(𝑥) (b) 2 cos(𝑥) (c) ln(3𝑥) (d) log10 (𝑥)
𝑥 𝑥 2
(e) −4𝑒 (f) 2𝑒 + 3𝑥 − tan(𝑥)
10. If 𝑦 = 2𝑥 3 − 11𝑥 2 + 12𝑥 − 5, determine:
𝑑𝑦 𝑑2 𝑦 𝑑𝑦
(a) and (b) the values of 𝑥 at which 𝑑𝑥 = 0 .
𝑑𝑥 𝑑𝑥 2
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