If y = e^{2x} \tan x, what is \frac{dy}{dx}? — JAMB UTME

If $y = e^{2x} \tan x$ , what is $\frac{dy}{dx}$ ?

e^{2x}(2\tan x + \sec^2 x)

Applying the product rule

\frac{d}{dx}[uv] = u'v + uv'

with

u = e^{2x}

and

v = \tan x

gives

u' = 2e^{2x}

and

v' = \sec^2 x

. Substituting these into the formula yields

2e^{2x}\tan x + e^{2x}\sec^2 x = e^{2x}(2\tan x + \sec^2 x)

Identify the component functions for the product rule

Let

u(x) = e^{2x}

and

v(x) = \tan x

The given function

y = e^{2x} \tan x

is a product of two functions.

Differentiate each component function separately

\frac{du}{dx} = 2e^{2x}

by the chain rule, and

\frac{dv}{dx} = \sec^2 x

The derivative of

e^{kx}

k e^{kx}

and the derivative of

\tan x

\sec^2 x

Apply the product rule formula

\frac{dy}{dx} = u \frac{dv}{dx} + v \frac{du}{dx}

\frac{dy}{dx} = e^{2x} \cdot \sec^2 x + \tan x \cdot 2e^{2x}

Combining the terms according to the standard product rule.

Factor out the common term

e^{2x}

\frac{dy}{dx} = e^{2x}(2\tan x + \sec^2 x)

Simplifying to match standard exam response format.

Differentiation of Exponential and Trigonometric Functions using the Product Rule

Estimated Time:1m 30s

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