would the derivative of f(x)=(7^x)(x^7) be f'(x)=(7^x ln7) (7x^6)?
no, you have to use the product rule
f(x) = (7^x)(x^7)
f ' (x) = (7^x)(7x^6) + (x^7)(ln7)(7^x)
= (7^x)(x^6) (7 + x ln7)
ok thanks!
Also, how did you simplify it to (7^x)(x^6) (7 + x ln7) ?
I took out the highest common factor
To find the derivative of the function f(x) = (7^x)(x^7), we can use the product rule of differentiation. The product rule states that the derivative of a product of two functions is equal to the derivative of the first function multiplied by the second function, plus the first function multiplied by the derivative of the second function.
Let's break down the function f(x) into two separate functions:
g(x) = 7^x
h(x) = x^7
Now, let's find the derivative of each function individually:
To find the derivative of g(x) = 7^x, we can use the chain rule. The chain rule states that the derivative of a function raised to a power is equal to the derivative of the exponential function multiplied by the original function raised to the power minus one, all multiplied by the derivative of the exponent.
g'(x) = (ln(7) * 7^x) * 1
= ln(7) * 7^x
To find the derivative of h(x) = x^7, we can use the power rule. The power rule states that the derivative of x raised to a constant power is equal to the constant multiplied by x raised to the power minus one.
h'(x) = 7 * x^(7-1)
= 7 * x^6
Now, using the product rule, we can find the derivative of f(x) = (7^x)(x^7):
f'(x) = g'(x) * h(x) + g(x) * h'(x)
= (ln(7) * 7^x) * (x^7) + (7^x) * (7 * x^6)
= (ln(7) * 7^x * x^7) + (7^(x+1) * x^6)
So, f'(x) = (ln(7) * 7^x * x^7) + (7^(x+1) * x^6).