Value after differentiating cos (x2+ 5) is ________a)5.sin (x2 + 5)b)-sin (x2 + 5).2xc)sin (x2 + 5).2xd)cos (x2 + 5).2xCorrect answer is option 'B'. Can you explain this answer?

Question

Answer

We differentiate the given function with help of chain rule and hence the outer function becomes –sin and the inner function is differentiated into 2x, therefore the answer comes out to be -sin (x² + 5).2x.

Answer

To find the value after differentiating the given expression, we need to apply the chain rule of differentiation.

The chain rule states that if we have a composite function, such as f(g(x)), where f and g are differentiable functions, then the derivative of the composite function is given by:

(f(g(x)))' = f'(g(x)) * g'(x)

In this case, our composite function is cos(x^2 - 5), where f(u) = cos(u) and g(x) = x^2 - 5.

Let's break down the steps to find the derivative:

Step 1: Find the derivative of the outer function f(u) = cos(u).
The derivative of cos(u) with respect to u is given by:
f'(u) = -sin(u)

Step 2: Find the derivative of the inner function g(x) = x^2 - 5.
The derivative of x^2 with respect to x is given by:
g'(x) = 2x
The derivative of -5 with respect to x is 0, as -5 is a constant.

Step 3: Apply the chain rule by multiplying the derivatives from steps 1 and 2.
(f(g(x)))' = f'(g(x)) * g'(x)
= -sin(g(x)) * 2x

Step 4: Substitute the inner function g(x) = x^2 - 5 back into the expression.
= -sin(x^2 - 5) * 2x

Therefore, the correct answer is option 'B': -sin(x^2 - 5) * 2x.

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