I need help with these two review calculus problems for my final exam.
What is the derivative of
a). (5x^2+9x-7)^5
b). Lne^x^3
For a. I got 50x+45(5x^5+9x-7)^4
For b. I got 3x^2
Can you verify please?
In google type:
wolfram alpha
When you see lis of results click on:
Wolfram Alpha:Computational Knowledge Engine
When page be open in rectangle type:
derivative (5x^2+9x-7)^5
and click option =
After few secons you will see result.
Then clic option Show steps
Then type :
derivative Ln(e^x^3)
and click option =
and show steps
To verify your answers, let's solve these calculus problems step by step.
a). To find the derivative of (5x^2+9x-7)^5, we can apply the chain rule.
The chain rule states that if we have a composition of functions, f(g(x)), then the derivative is given by f'(g(x)) multiplied by g'(x).
Applying the chain rule to this problem:
Let u = 5x^2 + 9x - 7
Let f(u) = u^5
We need to find f'(u) and u'.
f'(u) = 5u^4 (taking the derivative of u^5 with respect to u)
u' = 10x + 9 (taking the derivative of 5x^2 + 9x - 7 with respect to x)
Applying the chain rule, we have:
(f(g(x)))' = f'(g(x)) * g'(x)
= f'(u) * u'
= 5u^4 * (10x + 9) (substituting in the values of f'(u) and u')
= 5(5x^2 + 9x - 7)^4 * (10x + 9) (substituting in the value of u)
After simplifying, the final answer is:
= 5(5x^2 + 9x - 7)^4 * (10x + 9)
Since the derivative you obtained is slightly different, it seems there might have been an error in your answer.
b). To find the derivative of Lne^x^3, we can use the chain rule again.
Recall that the derivative of ln(u) with respect to u is 1/u.
In this case, let u = e^x^3 and f(u) = ln(u).
f'(u) = 1/u (taking the derivative of ln(u) with respect to u)
u' = 3x^2 * e^(x^3) (taking the derivative of e^x^3 with respect to x)
Applying the chain rule, we have:
(f(g(x)))' = f'(g(x)) * g'(x)
= f'(u) * u'
= (1/u) * (3x^2 * e^(x^3)) (substituting in the values of f'(u) and u')
= 3x^2 * e^(x^3) / (e^x^3) (substituting in the value of u)
= 3x^2
Your answer of 3x^2 is correct for the derivative of Lne^x^3.
Overall, the incorrect answer you obtained in part a seems to stem from a mistake during the application of the chain rule.