Question

How do you find the derivative of $y = {a^x}?$

Answer 1

Here, we find the first order of the derivative and by using the chain rule of the composite function. First of all we will apply the derivative of the given power in the function and then by using the chain rule will take the derivative of the function and then the derivative of the angle.

Complete step-by-step solution:
Take the given expression –
$y = {a^x}$ …. (A)
The above equation can be re-written as –
$y = {e^{\ln ({a^x})}}$ …. (B)
By using the law of the power of the logarithm,
$\Rightarrow y = {e^{x\ln (a)}}$
Take differentiation on both the sides of the above equations.
$\Rightarrow y' = \dfrac{d}{{dx}}({e^{x\ln (a)}})$
Applying the value for the differentiation in the above equation and applying the chain rule.
$\Rightarrow y' = ({e^{x\ln (a)}})\dfrac{{d(x\ln a)}}{{dx}}$
Place the value of the differentiation.
$\Rightarrow y' = ({e^{x\ln (a)}})\ln a$
Replace the values using the equations (A) and (B) in the above equation.
$\Rightarrow y' = {a^x}\ln a$
This is the required solution.

Additional Information: In mathematics, integration is the concept of calculus and it is the act of finding the integrals. One can find the instantaneous rate of change of the function at a point by finding the derivative of that function and placing it in the x-value of the point. Instantaneous rate of change of the function can be represented by the slope of the line, which says how much the function is increasing or decreasing as the x-values change.

Note: Know the difference between the differentiation and the integration and apply the formula accordingly. Differentiation can be represented as the rate of change of the function, whereas integration represents the sum of the function over the range. They are inverses of each other.