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Differentiate sums and differences

Differentiation is linear: for differentiable functions, the derivative of a sum is the sum of the derivatives, and the derivative of a difference is the difference of the derivatives, (f+g)′=f′+g′(f+g)'=f'+g' and (f−g)′=f′−g′(f-g)'=f'-g'. This allows a function’s rate of change or tangent slope to be found term by term, while preserving signs and recognizing that constant terms contribute zero; more advanced operator or abstract treatments are outside this scope.

Detailed Explanation: Differentiate sums and differences

To differentiate a sum or difference, differentiate each term separately and keep the signs unchanged:

(f+g)′=f′+g′and(f−g)′=f′−g′.(f+g)'=f'+g' \qquad\text{and}\qquad (f-g)'=f'-g'.

Remember:

  • Use the power rule: ddx(xn)=nxn−1\dfrac{d}{dx}(x^n)=nx^{n-1}.
  • The derivative of a constant is 00.

Example: Find the derivative of

y=3x4−5x2+7x−9.y=3x^4-5x^2+7x-9.

Differentiate each term one at a time:

ddx(3x4)=12x3,\frac{d}{dx}(3x^4)=12x^3, ddx(−5x2)=−10x,\frac{d}{dx}(-5x^2)=-10x, ddx(7x)=7,\frac{d}{dx}(7x)=7,

and

ddx(−9)=0.\frac{d}{dx}(-9)=0.

Now combine the results, preserving the original signs:

y′=12x3−10x+7.\boxed{y'=12x^3-10x+7}.

The constant term −9-9 disappears because constants do not change as xx changes.

Learn by doing: Differentiate sums and differences

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    Derivative Rules - Sum Rule Positive Powers to Derivative

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