How Do You Solve Calculus Problems Involving Rates of Change and Integration?

[studentxlol]

Homework Statement

Due to illness, I missed my calculus class and can't do the homework my friend gave me.

Q1) The volume of a sphere is increasing at a rate of 75cm^3s^-1. Find the rate at which the radius is increasing at the instant when the radius of the sphere is 15cm.

Q2) Find the gradient of the curve y=5e^3x at the point for which x=ln a, giving your answer in simplified form in terms of the constant a.

Homework Equations

The Attempt at a Solution

Q1) First idea was to use the chain rule?

Q2) y=5e^3x, dy/dx=15e^3x. Substitute x=ln a and rearrange to find a?

 

[Mark44]

Homework Statement

Due to illness, I missed my calculus class and can't do the homework my friend gave me.

Q1) The volume of a sphere is increasing at a rate of 75cm^3s^-1. Find the rate at which the radius is increasing at the instant when the radius of the sphere is 15cm.

Q2) Find the gradient of the curve y=5e^3x at the point for which x=ln a, giving your answer in simplified form in terms of the constant a.

Homework Equations

The Attempt at a Solution

Q1) First idea was to use the chain rule?

No, the first thing to do is to establish a relationship between the volume V and the radius r.
Then find a relationship between the rates (i.e., derivatives with respect to time) of those variables.

Q2) y=5e^3x, dy/dx=15e^3x. Substitute x=ln a and rearrange to find a?

Use parentheses around the exponents.

Let y = f(x) = 5e^(3x). Then f'(x) = 15e^(3x).
f'(lna) = ?
They are not asking you to find a.