# AP Calculus AB : Chain rule and implicit differentiation

## Example Questions

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### Example Question #1 : Chain Rule And Implicit Differentiation

.  Find the derivative.

Explanation:

When the function is a constant to the power of a function of x, the first step in chain rule is to rewrite f(x).  So, the first factor of f(x) will be .  Next, we have to take the derivative of the function that is the exponent, or .  Its derivative is 10x-7, so that is the next factor of our derivative.  Last, when a constant is the base of an exponential function, we must always take the natural log of that number in our derivative.  So, our final factor will be .  Thus, the derivative of the entire function will be all these factors multiplied together: .

### Example Question #2 : Chain Rule And Implicit Differentiation

Find the derivative of the function:

Explanation:

Whenever we have an exponential function with , the first term of our derivative will be that term repeated, without changing anything.  So, the first factor of the derivative will be .  Next, we use chain rule to take the derivative of the exponent.  Its derivative is .  So, the final answer is .

### Example Question #3 : Chain Rule And Implicit Differentiation

Find the derivative of the exponential function, .

Explanation:

To take the derivative of any exponential function, we repeat the exponential function in the derivative.  So, the first factor of the derivative will be .  Next, we have to take the derivative of the exponent using chain rule.  The derivative of the trigonometric function secx is secxtanx, so in terms of this problem its derivative is .  Since the angle has a scalar of 3, we must also multiply the entire derivative by 3.  So, the answer is .

### Example Question #4 : Chain Rule And Implicit Differentiation

Find the derivative of .

Explanation:

To find the derivative, we can first rewrite the function to make it easier to take the chain rule.  Rewrite  as .  Now, like in any exponential function, the first factor of the derivative is the original exponential function.  So, the first factor of f'(x) will be .  Next, by the chain rule for derivatives, we must take the derivative of the exponent, which is why we rewrote the exponent in a way that is easier to take the derivative of.  So, the derivative of the exponent is , because the 1/2 and the 2 cancel when we bring the power down front, and the exponent of 1/2 minus 1 becomes negative 1/2.  The last factor of the derivative is  because in every derivative of an exponential function where the base is a number, we must multiply by the natural log of that base.  So, once you multiply all these factors together, the final answer is

### Example Question #5 : Chain Rule And Implicit Differentiation

If  , find the derivative through implicit differentiation.

Explanation:

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  Next, factor out the y' from both terms on the left side of the equation so that we can solve for it:   To get y' alone, divide both sides by  to get .  To simplify even further, we can factor a 2 out of the numerator and denominator and cancel them.  So, the final answer is .

### Example Question #6 : Chain Rule And Implicit Differentiation

If , find .

Explanation:

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation and factor out a common y': .  To get y' alone, divide both sides by to get .

### Example Question #7 : Chain Rule And Implicit Differentiation

Find the derivative of the function of the circle

Explanation:

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get   The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  To get y' alone, divide both sides by  to get .  To simplify even further, we can factor a 2 out of the numerator and denominator and cancel them.  So, the final answer is .

### Example Question #8 : Chain Rule And Implicit Differentiation

Find the derivative of the function  using implicit differentiation.

cannot be solved

Explanation:

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get .  The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  To get y' alone, divide both sides by -3 to get .  To simplify even further, we can factor a -2 out of the numerator and denominator and cancel them.  So, the final answer is .

### Example Question #9 : Chain Rule And Implicit Differentiation

Find the derivative of the function .

Explanation:

To find the derivative through implicit differentiation, we have to take the derivative of every term with respect to x.  Don't forget that each time you take the derivative of a term containing y, you must multiply its derivative by y'.  So, when we take the derivative of each term, we get .  The next step is to solve for y', so we put all terms containing y' on the left side of the equation: .  Next, factor out the y' from both terms on the left side of the equation so that we can solve for it: .  To get y' alone, divide both sides by  to get a final answer of  .

### Example Question #10 : Chain Rule And Implicit Differentiation

Find the derivative of the function .

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