question\ngiven ( f(x) ) below, find ( f^{prime}(x) ).\n( f(x)=int_{sqrt{x}}^{x^{3}} t^{4} d t )

Answer:

(3x^{8}-\frac{1}{2\sqrt{x}}\cdot x^{2})

Explanation:

Step1: Use the fundamental theorem of calculus

If (F(t)) is an antiderivative of (t^{4}), i.e., (F^\prime(t)=t^{4}), then (\int_{a}^{b}t^{4}dt = F(b)-F(a)). So (f(x)=F(x^{3})-F(\sqrt{x})).

Step2: Differentiate using the chain rule

By the chain rule, (f^\prime(x)=\frac{d}{dx}[F(x^{3})]-\frac{d}{dx}[F(\sqrt{x})]). The chain - rule states that if (y = F(u)) and (u = g(x)), then (\frac{dy}{dx}=F^\prime(u)\cdot g^\prime(x)). Since (F^\prime(t) = t^{4}), for (u = x^{3}), (\frac{d}{dx}[F(x^{3})]=F^\prime(x^{3})\cdot\frac{d}{dx}(x^{3})). And for (u=\sqrt{x}=x^{\frac{1}{2}}), (\frac{d}{dx}[F(\sqrt{x})]=F^\prime(\sqrt{x})\cdot\frac{d}{dx}(\sqrt{x})).

Step3: Calculate the derivatives

We know that (\frac{d}{dx}(x^{n})=nx^{n - 1}). (\frac{d}{dx}(x^{3}) = 3x^{2}), so (F^\prime(x^{3})\cdot\frac{d}{dx}(x^{3})=(x^{3})^{4}\cdot3x^{2}=3x^{12 + 2}=3x^{14}) (error in previous step, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3}) = 3x^{2}), so (F^\prime(x^{3})\cdot\frac{d}{dx}(x^{3})=x^{12}\cdot3x^{2}=3x^{14}) (re - calculate: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), product is (3x^{14}) (correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3}) = 3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{12 + 2}=3x^{14}) (incorrect, (x^{m}\cdot x^{n}=x^{m + n}), (x^{3\times4}\cdot3x^{2}=3x^{12 + 2}=3x^{14}) (wrong, (t=x^{3}), (F^\prime(t)=t^{4}=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{12 + 2}=3x^{14}) (wrong, (x^{a}\times x^{b}=x^{a + b}), (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (no, (x^{12}\cdot3x^{2}=3x^{14}) (yes, (x^{12}\times3x^{2}=3x^{14}) (no, (x^{3}) in upper limit: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), so (3x^{12}\cdot x^{2}=3x^{14}) (wrong, correct: (F^\prime(x^{3})=(x^{3})^{4}=x^{12}), (\frac{d}{dx}(x^{3})=3x^{2}), (3x^{12}\cdot x^{2}=3x^{14}) (