DERIVATIVES OF EXPONENTIAL FUNCTIONS WORKSHEET

Question 1 :

Find the derivative of e^x with respect to x.

or

Given y = e^x, find dy/dx.

(x and y are variables and e is a constant)

Question 2 :

Find the derivative e^mx with respect to x, where m is a constant.

Question 3 :

Find the derivative e^-x with respect to x.

Question 4 :

Find dy/dx :

Question 5 :

Find dy/dx :

Question 6 :

Find the derivative e² with respect to x.

Question 7 :

Find the derivative of a^x with respect to x.

or

Given y = a^x, find dy/dx.

(x and y are variables and a is a constant)

Question 8 :

If y = 5^x, find dy/dx.

Question 9 :

Find the derivative of x^x with respect to x.

or

Find dy/dx, if y = x^x.

Question 10 :

Find the derivative of a^a with respect to x.

or

Given y = a^a, find dy/dx.

(a is a constant)

1. Answer :

In y = e^x, we have constant e in base and variable x in exponent.

y = e^x

Take logarithm on both sides.

lny = lne^x

Apply the power rule of logarithm on the right side.

lny = xlne

(The base of a natural logarithm is e, lne is a natural logarithm and its base is e)

lny = xln_ee

In a logarithm, if the base and argument are same, its value is 1. In ln_ee, the base and argument are same, so its value is 1.

lny = x(1)

lny = x

Find the derivative with respect to x.

(1/y)(dy/dx) = 1

Multiply both sides by y.

dy/dx = y

Substitute y = e^x.

dy/dx = e^x

Therefore, the derivative e^x is e^x.

Note :

In general, the derivative of e^x is e^x. That is, the derivative of e^variable is the same e^variable.

Using chain rule, we can explain the derivative of e^x.

That is, the derivative of e^x is e^x. So, far we have completed the derivative only for e^x, further, we have to find the derivative of the exponent x with respect to x, by chain rule.

y = e^x

dy/dx = e^x(1)

dy/dx = e^x

2. Answer :

The derivative of e^mx is e^mx, further, find the derivative of the exponent mx with respect to x. By chain rule. derivative of mx with respect to x is m(1) = m.

y = e^mx

dy/dx = e^x⋅ m(1)

dy/dx = me^x

So, the derivative e^mx with respect to x is me^x.

3. Answer :

The derivative of e^-x is e^-x, further, find the derivative of the exponent -x with respect to x. By chain rule. derivative of -x with respect to x is -1.

y = e^-x

dy/dx = e^-x⋅ (-1)

dy/dx = -e^-x

So, the derivative e^-x with respect to x is e^-x.

4. Answer :

5. Answer :

6. Answer :

The derivative of e² is e², further, find the derivative of the exponent 2 with respect to x. By chain rule. derivative of 2 with respect to x is 0.

y = e²

dy/dx = e²⋅ (0)

dy/dx = 0

So, the derivative e² with respect to x is zero.

Note :

We know that e is a mathematical constant and number 2 is a constant. So, e² is a constant. Since the derivative of a constant is zero, the derivative e² is zero.

Derivative of e to the power of any constant is zero.

7. Answer :

In y = a^x, we have constant a in base and variable x in exponent.

y = a^x

Take logarithm on both sides.

lny = lna^x

Apply the power rule of logarithm on the right side.

lny = xlna

Find the derivative with respect to x.

(1/y)(dy/dx) = (1)lna

(since, a is a constant, lna is also a constant. When we find derivative xlna, we keep the the constant lna as it is and find the derivative of x with respect to x, that is 1)

Multiply both sides by y.

dy/dx = ylna

Substitute y = a^x.

dy/dx = a^xlna

Therefore, the derivative a^x is a^xlna.

8. Answer :

y = 5^x

dy/dx = 5^xln5

9. Answer :

In y = x^x, we have variable x in exponent.

y = x^x

Take logarithm on both sides.

lny = lnx^x

Apply the power rule of logarithm on the right side.

lny = xlnx

Find the derivative with respect to x.

(1/y)(dy/dx) = x(1/x) + lnx(1)

(to find the derivative of xlnx, product rule is used)

(1/y)(dy/dx) = 1 + lnx

Multiply both sides by y.

dy/dx = y(1 + lnx)

Substitute y = x^x.

dy/dx = x^x(1 + lnx)

Therefore, the derivative x^x is x^x(1 + lnx).

10. Answer :

In y = a^a, In both exponent and base, we have the constant a.

Since a is a constant, a^a is also a constant.

y = a^a

dy/dx = 0

Since the derivative of a constant is zero, the derivative of a^a is zero.

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