Gujarat Board Statistics Class 12 GSEB Solutions Part 2 Chapter 5 Differentiation Ex 5.1 Textbook Exercise Questions and Answers.
Gujarat Board Textbook Solutions Class 12 Statistics Part 2 Chapter 5 Differentiation Ex 5.1
Obtain the derivatives of the following functions with the help of definition:
Question 1.
f(x) = 2x + 3
Solution:
Here, f(x) = 2x + 3
∴ f(x + h) = 2 (x + h) + 3 = 2x + 2h + 3
Hence, f(x) = 2x + 3 then f’(x) = 2.
f(x) = x2
Solution:
Here, f(x) = x2
∴ f(x + h) = (x + h)2 = x2 + 2xh + h2
Hence, f(x) = x2 then f'(x) = 2x.
Question 3.
f(x) = x7
Solution:
Here, f(x) = x7
∴ f(x + h) = (x + h)7
Take, x + h = t, when h → 0, t → x and h = t – x
Hence, f(x) = x7 then f'(x) = 7x6
Question 4.
f(x) = \frac{1}{x+1}, x ≠ -1
Solution:
Here, f(x) = \frac{1}{x+1}
∴ f(x + h) = f(x)
Hence, f(x) = \frac{1}{x+1} then f'(x) = -\frac{1}{(x+1)^{2}}.
Question 5.
f(x) = \sqrt[3]{x}
solution:
Here, f(x) = \sqrt[3]{x}=x^{\frac{1}{3}}
∴ f(x + h) = (x+h)^{\frac{1}{3}}
Take, x + h = t, when h → 0, t → x and h = t – x
Hence, f(x) = \sqrt[3]{x} then f'(x) = \frac{1}{3 \cdot x^{\frac{2}{3}}}.
Question 6.
f(x) = 24, x ≠ \frac{4}{3}
Solution:
Here, f(x) = \frac{2}{3 x-4}
∴ f(x + h) = \frac{2}{3(x+h)-4}
Hence, f(x) = \frac{2}{3(x+h)-4} then f’(x) = \frac{-6}{(3 x-4)^{2}}.
Question 7.
f(x) = 10
Solution:
Here, f(x) = 10
∴ f(x + h) = 10
Hence, f(x) = 10 then f’(x) = 0