differentiate arctan(7^x)?

Question

this very frustrating , this is what i did:

1
-------------- * (x7^(x-1))
1 + (7^x)^2

what am i missing? or did I approach it wrong?

why doesn't the power rule hold for 7^x?

Answer 1

u = 7^x
ln(u) = xln(7)
Use implicit differentiation

(1/u)du = ln(7)dx
du/dx = y.ln(7)
= (7^x)ln(7)

y= arctan(7^x)
= arctan v
dy/dv = 1/(1+v^2)
= 1/(1+7^(2x))
dv/dx = (7^x)ln(7)

dy/dx = (dy/dv)(dv/dx)
= (7^x)ln(7)/(1+7^(2x))
-----------------------------------------------

y=x^n
dy/dx = nx^(n-1) if and only if n is a constant

In this question the exponent is a variable (x) and so implicit differentiation is required.

Answer 2

Your answer is wrong. The derivative of x^7 is 7*x^6, but the derivative of 7^x is (7^x)*ln(7). So the correct answer to your problem is:

[(7^x)*ln(7)] / [1 + 7^(2x)]

Answer 3

You were very close, but the derivative of 7^x is (ln 7) * 7^x. Fortunately, it's easy to fix.

Answer 4

The derivative of 7^x is 7^x ln7 , so your answer should be
7^x ln7/(1+7^2x)