Differential calculus is one of sub field of calculus which is concerned with the study of rate at which quantity changes. The object of study of differential calculus is derivative of a function .Differentiation is a process of finding Derivative. Differential calculus is connected with fundamental theorem of calculus. Differentiation is used in many applications. For example, in physics the derivative of velocity with respect to time is acceleration. Derivative of a function f(x) is defined by d/ d x (f(x).
Problem 1: Compute the derivative of the function f(x) = 5x^3 - 9x +8cos x
Solution: Given the function f(x) = 5x^3 - 9x +8cos x
=> To find derivative of 5x^3 subtract one from the variable power and multiply the original power by the variable.
=> Here 3 is the power of x. subtract one from 3. (3-1 = 2)
=> Now multiply the original power by the variable, which gives 5. 3x^2 = 15x^2
=> Now take -9x, we get the derivative as -9
=> For 8 cos x = - 8 sin x (since cos x = - sin x)
=> Therefore, the derivative of 5x^3 - 9x +8cos x is 15x^2 -9 – 8 sin x
Problem 2: Compute the derivative of the function f(x) = (x^2 + 1) ^5
Solution: Given the function f(x) = (x^2 + 1) ^5
=> To find derivative of this function take (x^2 + 1) as y
=> So y^5 = 5y^4 (as per power rule of derivative)
=> Now find the derivative of y = x^2 + 1 = 2x (power rule)
=> Derivative d f(x) / d x= 5y^4. 2 x
=> We know y = (x^2 + 1)
=> Therefore, Derivative of f(x) = 5(x^2 + 1) ^4. 2x = 10x(x^2 + 1)^4.