evaluate the derivative of the following function.\nf(x)=5x sin^{-1}x\nf(x)=\\square
evaluate the derivative of the following function.\nf(x)=5x sin^{-1}x\nf(x)=\\square
Answer
Explanation:
Step1: Apply the product rule
The product rule states that if (y = u\cdot v), then (y^\prime=u^\prime v + uv^\prime). Let (u = 5x) and (v=\sin^{-1}x). First, find (u^\prime) and (v^\prime). (u^\prime=\frac{d(5x)}{dx}=5). The derivative of (y = \sin^{-1}x) is (y^\prime=\frac{1}{\sqrt{1 - x^{2}}}), so (v^\prime=\frac{1}{\sqrt{1 - x^{2}}}).
Step2: Substitute into the product rule formula
(f^\prime(x)=u^\prime v+uv^\prime) (f^\prime(x)=5\cdot\sin^{-1}x+5x\cdot\frac{1}{\sqrt{1 - x^{2}}})
Answer:
(5\sin^{-1}x+\frac{5x}{\sqrt{1 - x^{2}}})