f(x)=x^9h(x)
h(−1)=5
h′(−1)=8
Calculate f′(−1).
To calculate f'(−1), we need to find the derivative of the function f(x) = x^9h(x).
In order to do this, we can use the product rule for differentiation. The product rule states that if we have two functions u(x) and v(x), then the derivative of their product is given by:
(uv)' = u'v + uv'
Applying the product rule to our function f(x) = x^9h(x), we let u(x) = x^9 and v(x) = h(x). Thus, we have:
f'(x) = (x^9)'h(x) + x^9(h(x))'
Now, let's find the derivatives of the functions u(x) = x^9 and v(x) = h(x).
The derivative of u(x) = x^9 is given by the power rule, which states that the derivative of x^n is equal to n * x^(n-1). Applying the power rule to u(x) = x^9, we have:
u'(x) = 9x^(9-1) = 9x^8
For v(x) = h(x), we're given that h'(−1) = 8.
Plugging these derivatives back into the equation for f'(x), we have:
f'(x) = 9x^8 * h(x) + x^9 * h'(x)
Now, to find f'(−1), we substitute x = −1 into the equation:
f'(−1) = 9(−1)^8 * h(−1) + (−1)^9 * h'(-1)
First, let's simplify (−1)^8. Since any positive, even exponent of -1 is 1, we have:
(−1)^8 = 1
Similarly, (−1)^9 = −1.
Substituting these values back into the equation for f'(−1), we get:
f'(−1) = 9(1) * h(−1) + (−1) * h'(-1)
Given that h(−1) = 5 and h'(-1) = 8, we can substitute these values into the equation:
f'(−1) = 9(1) * 5 + (−1) * 8
= 45 + (−8)
= 37
Therefore, f'(−1) = 37.