Find the derivative.
f(x) = x2 · log7(x)
Give your answer using the form below.
xA(B + C logD(x))
To find the derivative of the function f(x) = x^2 * log7(x), we need to use the product rule and the chain rule.
The product rule states that for two functions u(x) and v(x), the derivative of their product is given by:
(uv)' = u'v + uv'
In this case, let's consider u(x) = x^2 and v(x) = log7(x).
First, let's find the derivative of u(x) = x^2:
u'(x) = 2x
Next, let's find the derivative of v(x) = log7(x). We need to use the chain rule for this.
The chain rule states that for a composition of functions f(g(x)), the derivative is given by:
(f(g(x)))' = f'(g(x)) * g'(x)
In this case, let's consider f(u) = log7(u) and g(x) = x.
First, let's find the derivative of f(u) = log7(u) with respect to u:
f'(u) = 1/u * log(7)
Next, let's find the derivative of g(x) = x:
g'(x) = 1
Now, using the chain rule, the derivative of v(x) = log7(x) is:
v'(x) = f'(g(x)) * g'(x) = (1/x) * log(7) * 1 = log(7)/x
Now, applying the product rule to u(x) = x^2 * log7(x), we have:
f'(x) = u'(x)v(x) + u(x)v'(x) = 2x * log7(x) + x^2 * (log(7)/x)
Simplifying the expression, we get:
f'(x) = 2x * log7(x) + x * log(7)
So, the derivative of f(x) = x^2 * log7(x) is given by:
f'(x) = x * (2 + log(7))