Calculate the derivatives of the following functions

Question

4x²e^5x

 

(t²+tan t)³

 

√x+2 cos³ x / 2x-5

 

sec(tan x)

 

(√x+(√x))

 

ln(3x²+5)

 

5^lnx

 

xsin^-1 (1-x²)

Answer 1

Part 1

 

4x²e^5x

 

We will use a combination of product rule and chain rule.  Let f(x) = 4x²    and g(x) = e^5x

 

Product rule is  f'(x)g(x) + f(x)g'(x).

Chain rule is the derivative of the inside function multiplied by the derivative of the outside function evaluated inside.

The derivative of an exponential function is the function itself.

 

Our derivative is

 

8xe^5x + 4x²5e^5x  =

 

8xe^5x + 20x²e^5x

 

Part 2

 

We will use chain rule for this one.

 

The derivative is

 

3(t² + tan t)² * (2t + sec²t)

 

 

Part 3

 

Use the quotient rule:

 

[f'(x)g(x) - (f(x)g'(x))] / g²(x)

 

You will also use the chain rule as needed.

 

 

Part 5

 

√(x) + √(x) = 2√(x) = 2(x)^1/2

 

The derivative is

 

2*(1/2)(x)^-1/2 =

 

x^-1/2 = 1/√(x)

 

 

Part 6

 

ln(3x² + 5)

 

The derivative of natural log function is the reciprocal of that function. 

 

6x / (3x² + 5)

 

 

Part 7

 

I think you meant 5lnx and not 5^lnx.

 

The derivative is

 

5*(1/x) = 5/x

 

 

Now that you understand the concepts, you can try out the other two.