Find the derivative of y=15x^7+7^x+22
105x^6 + ln7 7^x
Thank you
To find the derivative of the given function y = 15x^7 + 7^x + 22, we need to use the power rule and exponential function differentiation.
Let's differentiate each term separately:
1. The derivative of 15x^7 with respect to x:
Apply the power rule: d/dx [x^n] = n * x^(n-1)
In this case, n = 7, so the derivative is:
dy/dx = 7 * 15 * x^(7-1)
= 105x^6
2. The derivative of 7^x with respect to x:
Since 7^x consists of an exponential function, we need to use the chain rule.
The chain rule states that d/dx [f(g(x))] = f'(g(x)) * g'(x)
Let f(u) = 7^u and g(x) = x.
So, f'(u) = 7^u * ln(7), which is the derivative of 7^u, and g'(x) = 1, which is the derivative of x.
Applying the chain rule, the derivative is:
dy/dx = f'(g(x)) * g'(x)
= 7^x * ln(7) * 1
= ln(7) * 7^x
3. The derivative of the constant term 22 with respect to x is 0 since it does not depend on x.
Finally, combining the derivatives of all terms, we get:
dy/dx = 105x^6 + ln(7) * 7^x