Find first derivative of following function x^2 log4 (x+1)^2
To find the first derivative of the function f(x) = x^2 * log4((x+1)^2), we will use the product rule and the chain rule of differentiation.
Step 1: Apply the product rule
The product rule states that if we have two functions, u(x) and v(x), then the derivative of their product is given by: (u(x) * v'(x)) + (u'(x) * v(x)).
In this case, u(x) = x^2 and v(x) = log4((x+1)^2).
Now, we need to find the derivatives of u(x) and v(x).
The derivative of u(x) = x^2 is u'(x) = 2x.
Step 2: Apply the chain rule
To find the derivative of v(x) = log4((x+1)^2), we need to apply the chain rule. The chain rule states that if we have a composite function, g(f(x)), then the derivative of that function is given by: g'(f(x)) * f'(x).
In this case, g(x) = log4(x) and f(x) = (x+1)^2.
To find the derivative of g(x) = log4(x), we can use the fact that the derivative of log base a of x is given by: (1 / (x * ln(a))).
Therefore, the derivative of g(x) = log4(x) is g'(x) = (1 / (x * ln(4))).
To find the derivative of f(x) = (x+1)^2, we can apply the power rule, which states that if we have a function of the form g(x) = x^n, then the derivative of g(x) is given by: g'(x) = n * x^(n-1).
Therefore, the derivative of f(x) = (x+1)^2 is f'(x) = 2(x+1).
Step 3: Combine the derivatives from steps 1 and 2
Now that we have the derivatives of u(x) and v(x), we can apply the product rule to find the derivative of the entire function f(x) = x^2 * log4((x+1)^2).
Using the product rule formula, we have:
f'(x) = (u(x) * v'(x)) + (u'(x) * v(x))
Substituting the values we found earlier, we get:
f'(x) = (x^2 * (1 / (x * ln(4)))) + (2x * log4((x+1)^2))
Simplifying further, we get:
f'(x) = (1 / ln(4)) + 2x * log4((x+1)^2))
Therefore, the first derivative of the function f(x) = x^2 * log4((x+1)^2) is (1 / ln(4)) + 2x * log4((x+1)^2)).