Problem Statement: Differentiate with respect to x e^3 root ax ×sin(bx^2-c)

Solution:

We need to find the derivative of the given function with respect to x. Let's break it down and simplify it first.

e^3 root ax ×sin(bx^2-c) = e^(3√ax) × sin(bx^2-c)

Now, we can use the product rule of differentiation to find the derivative of the given function.

Product Rule: (uv)' = u'v + uv'

Here, u = e^(3√ax) and v = sin(bx^2-c)

Step 1: Find u'

Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x)

Let f(x) = e^x and g(x) = 3√ax

f'(x) = e^x and g'(x) = (1/2) * (a^(-2/3)) * x^(-1/3)

Therefore, u' = f'(g(x)) * g'(x) = e^(3√ax) * (1/2) * (a^(-2/3)) * x^(-1/3)

Step 2: Find v'

Chain Rule: (f(g(x)))' = f'(g(x)) * g'(x)

Let f(x) = sin(x) and g(x) = bx^2-c

f'(x) = cos(x) and g'(x) = 2bx

Therefore, v' = f'(g(x)) * g'(x) = cos(bx^2-c) * 2bx

Step 3: Find (uv)'

Now, we can use the product rule to find the derivative of the given function.

(uv)' = u'v + uv'

= e^(3√ax) * (1/2) * (a^(-2/3)) * x^(-1/3) * sin(bx^2-c) + e^(3√ax) * cos(bx^2-c) * 2bx

= e^(3√ax) * (1/2) * (a^(-2/3)) * x^(-1/3) * sin(bx^2-c) + 2bx * e^(3√ax) * cos(bx^2-c)

Therefore, the derivative of e^3 root ax ×sin(bx^2-c) with respect to x is e^(3√ax) * (1/2) * (a^(-2/3)) * x^(-1/3) * sin(bx^2-c) + 2bx * e^(3√ax) * cos(bx^2-c).