question 17 - 1 point given f(x) below, find f(x). f(x)=int_{2x}^{x^{2}}(t^{2}-5)^{6}dt provide your answer…

question 17 - 1 point given f(x) below, find f(x). f(x)=int_{2x}^{x^{2}}(t^{2}-5)^{6}dt provide your answer below: f(x)=

question 17 - 1 point given f(x) below, find f(x). f(x)=int_{2x}^{x^{2}}(t^{2}-5)^{6}dt provide your answer below: f(x)=

Answer

Explanation:

Step1: Apply the fundamental theorem of calculus and chain - rule

If (F(t)) is an antiderivative of (g(t)=(t^{2}-5)^{6}), i.e., (F^\prime(t)=g(t)), then (f(x)=F(x^{2})-F(2x)).

Step2: Differentiate (f(x)) using the chain - rule

By the chain - rule, (\frac{d}{dx}F(u)=F^\prime(u)\cdot u^\prime). For (y = F(x^{2})), let (u = x^{2}), then (\frac{d}{dx}F(x^{2})=F^\prime(x^{2})\cdot2x). For (y = F(2x)), let (u = 2x), then (\frac{d}{dx}F(2x)=F^\prime(2x)\cdot2). Since (F^\prime(t)=(t^{2}-5)^{6}), we have (f^\prime(x)=(x^{2})^{2}-5)^{6}\cdot2x-( (2x)^{2}-5)^{6}\cdot2).

Step3: Simplify the expression

(f^\prime(x)=2x(x^{4}-5)^{6}-2(4x^{2}-5)^{6})

Answer:

(f^\prime(x)=2x(x^{4}-5)^{6}-2(4x^{2}-5)^{6})